Angra Mainyu's blog

The KCA is defended by a some theist philosophers in combination with other arguments intended to bridge the gap between the conclusion that the universe has a cause and a conclusion that the cause of the universe is a being with certain properties, in particular a personal agent of great power.

I will call the combination of the KCA with those other arguments, "KCA+".

In this essay, I will assess the KCA+, and argue that it provides no support for the conclusion that there is a powerful personal agent who created the universe, both by challenging the premises of the KCA itself, and the creation account proposed in usual versions of the KCA+.

b. I will focus mostly on William Lane Craig's version of the KCA+, but I will also assess some alternative arguments.

c. I will use the word 'argument' loosely, to denote (depending on context) either formal arguments (i.e., premises and conclusion), or the arguments (in the sense of 'arguing a case') given in defense of said premises; these usages are common and shouldn't cause confusion, given context.

d. I will call the parts of this essay 'sections' and 'subsections', without using "sub-subsections" or similar terms. Context and if necessary link to different parts of the essay should prevent ambiguity.

e. By 'incoherent' I mean that a claim is either meaningless, or meaningful but contradictory.

2. An analysis of the proposed creator, and the proposed creation account.

According to William Lane Craig, the creator proposed by the KCA+ is timeless without creation, and temporal with creation. But what does the word 'timeless' mean, when applied to an agent? Is such claim coherent? Is it compatible with the premises of the KCA?

In this section I will assess those issues, and generally the issues of the meaning and coherence of Craig's creation account.

2.1. Timelessness.

The word 'timeless' is not a word colloquially used, in the relevant sense.[2]

Yet, Craig does not seem to provide a clear definition, which leaves the matter of the meaning of the claim that God is timeless sans creation but temporal with it, obscure at best.

In order to try to ascertain what he means by that, one potential approach would be to try to grasp the meaning of 'timeless' by looking at a list of things generally regarded as 'timeless' by those philosophers who use such term, and a list of things that are no so regarded.

So, on the first list, we would have the number 5, the set of natural numbers, the proposition that water is H2O, etc.; in the second, we would have trains, planets, humans, mosquitoes, viruses, laptop computers, etc.

But then, based on those lists, when someone claims that a personal being is timeless – or, for that matter, timeless sans creation -, it becomes clear that, at least for me, that approach is not working, since either the claim is obviously false, or I've not been able to grasp the meaning of 'timeless'.

Also, if I were to speculate, my impression from those lists would be that 'timeless' is an element in some abstract domain with perhaps some particular properties, but clearly God isn't that.

Perhaps, given Craig's relational theory of time, it might be that being timeless is equivalent to not standing in temporal relationships. But that does not seem to help, either.

How can one make sense of a claim that an agent does not stand in temporal relationships sans his creation, but does stand in temporal relations with creation?

It's puzzling. But regardless, let's assess some of Craig's claims on the issues of creation, timelessness, changelessness, etc., and generally his descriptions of creation in greater detail.

2.1.2. Timelessness and change.

Craig claims that in his timeless state, God is changeless, or unchanging. However, Craig also claims that God can change and changed and ceased to be timeless.[3 ]

But how can we make sense of those claims?

We can understand the idea that an object remains changeless for a while and then changes; in other words, we can understand that an object remains the same as time goes by, during some period, etc., and then no longer stays the same.

However, clearly that is not applicable here, since that would imply that God's changeless state obtains during a temporal interval through which God remains unchanged, yet Craig is claiming timelessness in addition to changelessness, and while Craig's usage of the word 'timeless' is not clear, it is at least clear that a timeless state cannot be one during which time goes by.

So, the claim of changelessness is also very obscure.

Moreover, in the context of his defense of the KCA, Craig claims that instantaneous changes are impossible, and that every change has a finite, non-zero duration. [4]

So, let's consider the change in God from being timeless to being temporal. That is a change, and so it follows from Craig's claims that it has a finite, non-zero duration, say d > 0.

Thus, there is a temporal interval of non-zero duration d that ends at the first temporal state in the world, namely the first state at which God is temporal. That entails there was a temporal interval of non-zero duration d before there was time. But that is absurd.

Craig addressed that objection or an essentially similar one[5 ], contending that:

a. When he defines what "event" means[4], in the context of his defense of the KCA, he's trying to rule out instantaneous changes, and in that technical sense of 'event', anything that is instantaneous would not count as an event.

b. The creation of the universe by God or, for that matter, God's changing from timeless to temporal, would not qualify as an event in that sense, since it's instantaneous.

c. When he said that God changed and ceased to be timeless, he was using the word 'change' in a different sense from the sense in which he was using that word in the context of the Kalam Cosmological Argument. Furthermore, when he said that God changed, in that context, he merely meant that God's properties at his timeless state are different from his properties at his first temporal state. [5 ]

d. In any case, any problem can be resolved simply by stipulating, in the context of the defense of the KCA, that one is talking about changes of equal, non-zero and arbitrary duration, so the alleged contradiction can be resolved simply by a clarification of one's terms. [5 ]

However, all four claims a. - d. above are erroneous, for the following reasons:

a. Regardless of what he intended to do, Craig defined 'event' to mean 'any change', and then claimed that because any change takes time, then under that definition, there can't be any instantaneous events. But it's not that anything instantaneous would not count as an event by the definition of 'event'. Rather, the conclusion that anything instantaneous would not be an event follows from the definition of 'event' plus Craig's claim that any change takes time. [4]

b. Actually, since 'event' means 'any change' in the sense in which Craig uses the term 'event' in the context of his defense of the KCA, God's becoming temporal does qualify as an event, since that is clearly a change in God. If said change in God is instantaneous and has no finite, non-zero duration, then that contradicts Craig's claim that any change takes time.

c. If by saying that God changed and ceased to be timeless, Craig had merely meant that God's properties at his timeless state are different from the properties God has at his first temporal state, then for that matter Craig might as well had said that God changed and ceased to be temporal, becoming timeless. But clearly, in context, that would be a very different claim. So, considering context, one can ascertain that Craig didn't merely meant that God has different properties at those two states, but precisely that...God changed, using the word 'change' in the usual sense.

The usual sense of 'change' appears to be the way in which Craig used 'change' in the context of the defense of the KCA as well[4], since he defined 'event' in terms of change, but gave no definition of 'change' or made any suggestion that he was using 'change' in a non-standard, technical fashion.

d. Stipulating that one is going to talk about those events in particular does not change the fact that the claim that any change takes time and has a finite, non-zero duration, plus the claim or implication that God changed from a timeless state to a first temporal state, entails a contradiction. The contradiction still follows from Craig's claims, as he stated them.

In my assessment, Craig's contradictory claims plus the fact that in his reply he misunderstood the claims he had made earlier, only compound the problem of the obscurity of his creation account, raising doubts about his own understanding of the position he's defending, and the coherence of said position.

Still, we may consider whether there is a way out for a defender of a position similar to Craig's.

One way out of this particular difficulty would be to deny that any change takes a positive amount of time. In fact, there seems to be no other way out, since time before time makes no sense, and God clearly changes, on Craig's description or similar ones, from a state at which he's timeless, to a state at which he is not timeless.

However, denying that any change takes a positive amount of time would raise another difficulty, namely that of potentially instantaneous changes under a theory of time that maintains that time is discrete, which is the kind of theory defended by Craig and usually espoused by defenders of the KCA+.

More precisely, if time is dense, and – for instance – there are two temporal intervals A=[a, b) and B=[b, c), then the temporal distance between A and B is zero. If P obtains all through A, and ¬P obtains all through B, there is a sense in which it may make sense to say that the change from P's obtaining to P's not obtaining is instantaneous. Yet, if time is discrete, this option is not available, and it's hard to see how one can make sense of instantaneous changes in discrete time.

Perhaps, someone who defends the KCA+ might say that all changes require a positive amount of time except for the change in God from timeless to temporal, the change from a state without creation to a state with creation, etc. However, that would look like special pleading, unless the person making the claim provides good reasons for the exception for the first change.

2.1.3. Timelessness, changelessness and quiescence.

According to Craig, the universe cannot have existed in an "absolutely quiescent state", like God's initial timeless state. [4]

However, as in the case of changelessness, that raises the question of how we can make sense of such a claim[7 ]; usually, we would understand that a quiescent object is one that is at rest as time goes by, but given Craig's claim of timelessness, the claim of quiescence is puzzling.

In the rest of this subsection, I will continue to analyze that claim, as well as the claim of changelessness, in the context of his creation account.

2.1.3.1. A brief analysis.

Let's consider first two scenarios, ordering states of the world causally.

Scenario a:

First state of the world:

Timeless state S. The only object that exists is O. [6]

Second state of the world:

Temporal state. t=0. The object are O and U, and O is the cause of the existence of U.

Third state of the world:

Temporal state; t=r > 0. The objects are O, U and, perhaps some other objects.

Scenario b:

First state of the world:

Initial temporal state; t=0. The only object is O.

Second state of the world:

Temporal state; t=r>0. The objects are O and U, and O is the cause of the existence of U.

Third state of the world:

Temporal state; t=s > r. The objects are O, U and, perhaps some other objects.

So, allegedly and going by Craig's statements, the [causally] first state of the world would be changeless and/or absolutely quiescent in scenario a, but neither changeless nor quiescent in scenario b. [7 ]

But that seems clearly false, since the first scenario and the second seem to be exactly the same in terms of changes and lack of quiescence. In particular, the entity O at the first stage of the world changes from that state to the second state of the world in the two scenarios.

Moreover, if, in scenario b, object O begins to exist, then it seems so would would object O in scenario a, and so in particular, Craig's God would be an entity that begins to exist but has no cause of his existence, contradicting premise 1 of the KCA.

On the other hand, if it is not the case that in scenario b, object O begins to exists, then for that matter it might be, for all we know, that if the past is finite, it is not the case that the universe begins to exist, contradicting premise 2 of the KCA.

In brief, those 'two' scenarios appear to be ontologically identical, despite the label 'timeless' in the first one.

At this point, someone might suggest that scenario a might not properly represent Craig's view of God's being timeless without creation and temporal with it, and that perhaps, on his view, there might not be a first temporal state at t=0, but instead an open temporal interval of the form (0, u] for some u>0, and that there are times arbitrarily close to zero, but not an initial moment. However, Craig's position entails that such open interval is impossible because it would be an actual infinity, so this objection would fail.

In the following subsections, I will analyze the matter in greater detail, considering other potential objections, and generally analyzing Craig's account in much greater detail.

2.1.3.2. Some terminology and assumptions for the rest of subsectio n 2.1.3.

1. I will use use bold italic uppercase letters (e.g., J, K, etc.) to denote finite temporal intervals, and bold italic lowercase letters (e.g., t, u, etc.) to denote temporal instants. I will also number them (e.g., t(1), t(2), etc.).

2. When I talk of temporal intervals or instants, I'm talking about actual features of the world – whatever those happen to be -, not about mathematical models.

3. When I speak of temporal intervals, I'm assuming those intervals are of finite duration, though not necessarily of equal duration.

4. By '◄' I mean 'earlier than', in the usual, temporal sense of 'earlier'. For instance, J◄K means that J and K do not overlap, and J happens earlier than K.

5. When I prove lemmas, unless otherwise stated, I'm assuming any parts of Craig's position, as required. In other words, those lemmas are statements that follow from Craig's position. I will explain why that is his position when such explanation is needed.

2.1.3.3. Changes and infinity.

Lemma 1: Infinitely many actual changes in the world are impossible, regardless of whether their duration is equal.

Proof:

This can easily be seen in Craig's defense of the KCA. For example, he claims that an infinite temporal regress is an actual infinity. [8 ]In context, Craig is talking about changes of equal temporal length, but clearly, his claim that they constitute an actual infinity in the sense of cardinality does not depend on their length.

2.1.3.4. Discrete time

Definition: A temporal interval J has property Q if and only if there is some positive integer N(J), such that J does not have more than N(J) temporal subintervals different from each other. Equivalently, J has property Q if and only if it is not the case that there are more actual different temporal subintervals of J than any natural number.

Lemma 2: If there are temporal intervals, then every temporal interval J has property Q.

Proof:

Let's assume there is an actual temporal interval J that does not have property Q.

By lemma 1, there are only finitely many actual changes that happen in the world during J. So, let m > 2 be a natural number greater than the number of actual changes during J.

Since J does not have property Q, there is some natural number r > (2* m) (for instance), such that J actually has r different temporal subintervals, J(1), J(2), J(3).., J(r).

Let us consider the following changes, for k in {1,...,r-1}

E(k): God changes from not knowing that J(k) is past to knowing that J(k) is past.

F(k): God changes from not knowing that J(k) is present to knowing that J(k) is present.

Note that those are actual changes in the world, more precisely in the mind of God. Given that the intervals are different from each other, there are at least r-1 > m distinct changes, contradicting the assumption. That proves lemma 2.

Lemma 3: If there are temporal intervals, every such interval J is divided in finitely many ordered temporal intervals, each of which has no actual proper subintervals.

Proof:

Let n(J) be the total number of actual temporal subintervals of each subinterval J; by lemma 2. n(J) is finite for all J.

Then, given some interval K, if n(K)=1, we're through, since K has no proper subintervals.

Suppose that lemma 3 is true for any J such that n(J) < n; n > 1, and let K be such that n(K)=n.

Then, let K(1) be a proper subinterval of K. Then, by hypothesis, the lemma applies both to K(1) and to any part of K before and after K(1). Then, by induction, the lemma is proved.

Lemma 4: If there are temporal instants, then for every two instants t and u such that t ◄ u, there is some non-negative integer N(t, u), such that there are no more than N(t, u) instants between t and u.

Proof:

Similar to the proof of lemma 2, with the required changes to address instants instead of intervals.

Lemma 5: Either there are temporal intervals that do not have actual divisions (i.e., no actual proper subintervals), or there are instants.

Proof:

On a tensed theory of time, temporal becoming is a real feature of the world, so if there are no intervals, it seems clear that there are instants. If there are intervals, there are intervals that don't have actual divisions in smaller intervals, by lemma 3.

2.1.3.5. An absolute temporal beginning

Lemma 6: If there are temporal intervals, there is an absolutely first such temporal interval K(0), which has no actual proper subintervals. Moreover, if there is more than one interval, then there is a second such interval K(1) with no proper subintervals, and generally the past is composed of finitely many such intervals.

Proof: Let's assume there are infinitely many past intervals {K(n)}, for all natural numbers n.

Let's consider the events

G(n): God changes from not knowing that K(n) is past to knowing that K(n) is past.

H(n): God changes from not knowing that K(n) is present to knowing that K(n) is present.

Then, there are infinitely many actual past changes, contradicting lemma 1.

That proves that if there are temporal intervals, there are finitely many past ones. Since each such interval is divided in finitely many subintervals each of which has no proper subintervals by previous lemmas, then it follows that there are finitely many intervals with no proper subdivisions, and a first such interval K(0).

If there is some other interval J such that begins earlier than J than K(0), then by lemma 3 we can divide J in intervals that are not further divided, and one of them would be earlier than K(0), a contradiction.

By a similar procedure, we can construct a second one, and so on, and the total number up to the present time must be finite by lemma 1, since otherwise we could construct an infinite sequence of temporal changes.

That proves lemma 6.

Lemma 7: If there are temporal instants, there is a first temporal instant t(0). Similarly, there is a second instant t(1), a third one t(2), and generally the past is composed of a finite number of such instants.

Proof:

Like the proof of lemma 6, removing superfluous parts and making the necessary changes to address instants instead of intervals.

Lemma 8: Either there is a first temporal instant t(0), or a first interval K(0) that has no proper temporal subintervals.

Proof:

It follows from lemmas 5, 6, and 7.

Lemma 9: There is an absolute first, temporal state of the world T(0), with no temporally proper states contained in it. Moreover, there is a second such state T(1), a third T(2), and generally the past if composed of finitely many such states.

If there are instants, then let T(n) be the state of the world at t(n), for all different past instants, taking t(n) ◄ t(n+1). It's clear, then, for k>n, T(k) is different from T(n), though we may as well prove that too, as follows: For k>n, at T(k) God knows that t(n) is past and that there are at least n past instants, whereas at T(n), it is not the case that God knows that t(n) is past and it is not the case that God knows that there are at least n past instants. So, if k>n, then the states T(k) and T(n) are actually different from each other. Also, since each T(n) corresponds to a single instant, it has no proper temporal subdivisions.

If there are no instants, then there are intervals with no proper subintervals by lemma 5. So, let T(n) be the state of the world at K(n), for all past intervals with no proper temporal subintervals, and taking K(n) ◄ K(n+1).

Then, for k>n, at T(k) God knows that K(n) is past and that there are at least n past intervals with no proper subintervals, whereas at T(n), it is not the case that God knows that K(n) is past and it is not the case that God knows that there are at least n past intervals with no proper subintervals. So, if k>n, then the states of the world T(k) and T(n) are different from each other.

Also, since each T(n) corresponds to a single interval with no proper subintervals, there are no actual temporal divisions in each state T(n).

2.1.3.6. No ontological difference

Let's consider now the following two scenarios, ordering the states in terms of causal priority.

Scenario 1:

First state of the world:

Timeless state S.

The only object is God, which exists timelessly at S and without a cause of his existence.

Second state of the world:

First temporal state T(0).

God exists temporally, and the universe[9 ] exists. The change from the first to the second state of the world, including the creation of the universe and God's own change from timeless to temporal, takes place because God intends to bring it about.

Third state of the world:

Second temporal state T(1). The objects are God, the universe, and perhaps some other objects.

Scenario 2:

First state of the world:

First temporal state T(0).

The only object is God, who exists temporally at T(0) and without a cause.

Second state of the world:

Second temporal state T(1).

God exists temporally, and the universe exists. The change from the first to the second state of the world, including the creation of the universe, takes place because God intends to bring it about.

Third state of the world:

Third temporal state T(2). The objects are God, the universe, and the same other objects as in scenario 1 (if any), with the same causes.

It appears that even though the word 'timeless' is used in the first scenario, the first causal state in scenario 1 is ontologically no different like the first temporal and causal state T(0) in scenario 2, since:

i. In both cases, what we have is a first state of the world changing into the next, then the next, and so on.

ii. In both cases, God exists without a cause of his existence at the first state of the world.

iii. In both cases, the universe exists at the second state of the world, and exists because God intends to bring it about, and so on.

iv. Generally, one is not able to find a distinction on those states by inspection of the description, other than the difference in the words that are used.

Also, saying that the difference between the two scenarios is that one state – namely, state S in scenario 1 – is timeless and changeless and the other – namely, T(0) in scenario 2 – is temporal and changing, or that in one of them there at least one tensed fact, whereas in the other one there are no tensed facts, etc., would fail to address the difficulty, since that would just amount to denying that there is no ontological difference, without explaining how or why that is so, which is a burden on the defender of the KCA +, since it seems intuitively very plausible that the states are ontologically identical just by looking at the description of those states, wording notwithstanding.

In particular, one can tell that despite the claim that the first state in scenario 1 is said to be timeless, it is not changeless. In fact, 'both' states are not changeless, and for the same reason, namely that just as God changes from his first to his second state in scenario 2, he does so in scenario 1.

So, it remains the case that there is an entity – i.e., God – that exists at a first state of the world and without a cause of his existence, changes to a second state, then to a third one, etc., regardless of whether the first state of that entity is called 'timeless'.

Thus, if the concept of timelessness is coherent and entails changelessness, then scenario 1 is improperly described as timeless, since the first state is not changeless, and hence not timeless. It appears, then, that scenario 1 is just scenario 2, plus a false claim of timelessness.

Let's consider now potential objections, claiming ontological differences between scenario 1 and scenario 2.

Objection 1: An ontological difference between scenario 1 and scenario 2 is that temporal states of God must change as time goes by, whereas the timeless state S in scenario 1 could have remained unchanged.

In other words, if God exists at temporal state T(0) – as in scenario 2 –, then necessarily, there will be a second temporal state T(1), and necessarily, God will change from T(0) to T(1). On the other hand, if God exists in a timeless state S (as in scenario 1), then it is possible, given that first timeless state S of God, that God never changes.

The problem with that objection is that it follows from any creation hypotheses posited by defenders of the KCA + – i.e., given what God's first state actually would have been, based on the implications of their claims –, that it is impossible that God never changes given that particular first allegedly timeless state S, which shows both scenario 1 and scenario 2 are indistinguishable in that regard as well, as the following reasoning shows (as usual, the states of the world are ordered causally):

Scenario 3:

First state of the world:

Timeless state S. The only object is God, which exists timelessly at S and without a cause.

Second state of the world:

Temporal state T(0). God exists temporally, and the universe exists temporally. The change from the first to the second state of the world, including the creation of the universe and God's own change from timeless to temporal, takes place because God intends to bring it about.

Scenario 4:

First state of the world:

Timeless state S'. The only object is God, which exists timelessly at S' and without a cause.

There are no temporal states of the world.

In scenario 3, if God does not intend at S to change, then at T(0) God just found himself altered.

In other words, if God's intent to change does not exist at S, then the change is not something brought about because God intends to bring it about, but something that happened to God, beyond his intent.

That is so because the change under consideration is a change in God's state, and from the causally first state S. In other words, it's a change from S to T(0).

So, if God does not intend at S to bring about the change, then said change cannot have been caused by God's intent at T(0), given that S is causally prior to T(0).

However, that contradicts the hypothesis that the change in scenario 3 happens because God intends to bring it about, rather than being something that happens to God.

Thus, at S, God intends to bring about the change. Hence, given God's state at S in scenario 3, it is impossible that God does not change, since God can't fail to bring about what he intends to bring about.

Moreover, just as in scenario 3 it is impossible that God does not change, the same is true and for the same reasons in scenario 1. Also, the state S in scenario 3 is different from the state S' in scenario 4, in which God does not change.

So, if the concept of timelessness is coherent and state S' in scenario 4 is a timeless state, then that state is ontologically different from any initial state of the actual world posited by defenders of the KCA+, which – despite claims of timelessness – remains ontologically indistinguishable from a first temporal state, as far as one can tell.

At this point, someone might suggest that, in scenario 3, God exercised his libertarian free will at T(0) to bring about the change, but at S, it is not the case that he intends to change.

However, leaving aside issues about the coherence of libertarian free will, that is impossible for the reasons I gave above, namely that since the change under consideration is God's change from S to T(0), it cannot have been decided at T(0), due to S's causal priority.

Thus, as shown above, at S God intends to change, and given S, God necessarily changes.

If that conclusion is incompatible with God's having libertarian free will at S, then that would not block the conclusion I proved above – since the argument goes through just as well -, so that would only mean that Craig's description is incompatible with God's having libertarian free will at S, creating a new problem for Craig's account, rather than resolving one.

On the other hand, if that conclusion is compatible with God's having libertarian free will at S, the point remains that given S, God necessarily changes, and so objection 1 fails.

Objection 2. An ontological difference between scenario 1 and scenario 2 is that in the case of a first temporal state of God in scenario 2, it is not only necessary but causally necessary that the state changes, whereas given timeless state S in scenario 1, it is necessary that God changes, but not causally necessary.

Actually, even if sometimes a state of affairs can determine but not causally determine following states, this is not the case of the state S and God's being a different state later, since given S, it is necessary and causally necessary that God changes.

The reason for that is the same given above, namely the change in God from the timeless state S to the first temporal state T(0) is a change from the causally first state of the world. So, given that what causes God to change from the first state S is that God intends to change, and nothing at T(0) or generally causally later than S could have caused a change from S, then God intends to change at S, and that is causally sufficient to bring about that God changes.

Objection 3. While it's true that there is no ontological difference between scenario 1 and scenario 2, the initial state in both scenarios is timeless, not temporal. The scenario improperly described is scenario 2, rather than scenario 1.

Assuming for the sake of the argument that the first state of God is somehow is properly called 'timeless' in scenario 1 for some reason, then it seems that the same is true of the first state of the universe in the following scenario, understanding that metaphysical time begins with the universe.

Scenario 5:

First state of the world:

The universe exists without a cause of its existence. Nothing else exists – there may or may not be some things in the universe, but nothing beyond the universe.

Second state of the world:

The universe has changed and exists in its second state. Nothing else exists.

Just as God exists without a cause in scenario1, scenario 2, and scenario 3, the universe[9 ] exists without a cause in scenario 5.

Just as God changes from the first to the second state of the world in scenario 1, scenario 2, and scenario 3, the universe so changes in scenario 5.

After that, God in those scenarios continues to change from one state of the world to the next, and the same is true of the universe in scenario 5.

Also, when the universe changes from its first state at the first state of the world to its second state at the second state of the world, there is no previous change, and no period during which the universe remains unchanged. But that is exactly what happens in the case of God in scenario 1.

Given the above, there seems to be no sense of 'quiescent' or 'changeless' in which the universe would fail to be quiescent or changeless in scenario 5, but God wouldn't fail to be so in scenario1.

So, there seems to be no ontological difference that would justify denying that the first state of God is temporal in scenario1, but wouldn't justify denying that the first state of the universe is temporal in scenario 5.

Someone might suggest that in scenario1 and scenario 2, the change from the first to the second state of the world, including the creation of the universe and God's own change from timeless to temporal, takes place because God intends to bring it about, whereas in scenario 5, the cause of the change is not specified.

However, that would be unrelated to the issue of whether God or the universe are 'timeless', changeless, quiescent, etc., and in any case, we may as well further specify the scenario and add that, in scenario 5, the change from the first to the second state of the universe is brought about by the universe's conditions at its first state (including, of course, any particles if there are any, etc., and the universe's causal powers and/or causal powers of the particles, etc.), regardless of whether we put that in terms of substances and causal powers, or in terms of substances and laws, etc.

Thus, the universe, which has certain conditions in its first state, brings about the change to its second state, just as God and his intent at the first state brings about the change in scenario1.

We may also stipulate if we so choose – though we don't need to – that the change indeterministic in scenario 5, so that given the first state of the universe, there is more than one possible second state, but only one actually obtains; or we may alternatively make it deterministic, etc.

In any case, that would not make any difference in terms of changelessness or quiescence.

2.1.4. Timelessness sans creation. More on Craig's description.

In this subsection, I will address a more elaborate description of God's alleged timelessness sans creation, which Craig gives in one of his articles[10].

In that article too, he claims that sans creation, God is changeless.

However, as I argued above, that claim would be false. More precisely, in the account of creation that Craig gives, God is not changeless, but changes just as he or the universe would change in scenarios in which there is no allegedly timeless state of affairs.

In addition, Craig claims that God is immobile. How can one make sense of such a claim?

It makes sense, of course, to say that an object remains immobile for a while, so as time goes by, the object does not move with respect to some frame of reference. But here the claim does not seem to be about spatial movement, so plausibly the claim is equivalent to the claim of changelessness, which has the shortcomings already addressed.

Else, Craig would have to explain what he means.

That aside, Craig gives an argument in support of the view that in his changeless state, God is also timeless. A first and decisive problem is that, as argued above, the state in question – i.e., the first state of God – would not be changeless. But let's consider Craig's argument in greater detail:

He claims that there is a possible world – say, W' - in which God refrains from creating anything, and claims that in that case, the state in question can be plausibly and coherently conceived of as "timeless".

Then, he argues that similarly, the first state of God in the actual world is timeless, since there is no intrinsic difference between the first state at the actual world and the first state at W'; Craig also claims that the initial segments of the actual world TW and the other world W' would be identical.

It's still not clear to me what Craig means by 'timeless' - if anything -, but leaving that aside, the claim that there is no intrinsic difference between the two states is false, at least in any relevant sense of 'intrinsic', for the following reasons:

Let S be the first state of the actual world TW, according to Craig's description. Then, God exists at S, and nothing else exists at S. Let S' be the only state of the world W', at which God exists alone.

Then, given S, necessarily God changes, as the analysis of scenario 3 above shows.

On the other hand, given S', it is not the case that necessarily, God changes, as the analysis of scenario 4 above shows. Furthermore, given S', God necessarily does not change, though showing that given S', it's not the case that God necessarily changes, is sufficient to show a clear ontological difference.

Similar considerations apply to causal necessity, also as argued above.

As for the claim that the initial segments would be identical, it is also false, since any initial segment of TW on Craig's account would contain the first state S, which is a state at which God and only God exists, and such that given such state S, necessarily God changes, whereas that is not true of any initial segment of W'.

So, if one assumes Craig's account of creation, then given any initial segment of the actual world, necessarily change occurs. Whether which changes occur are determined by such initial state is another matter, but there is no need to discuss it in this context.

In light of the fact that given any initial segment of the actual world assuming Craig's account of creation, change necessarily follows, whereas it's not the case that given any initial segment of W', change necessarily follows, we ought to conclude that Craig's claim is not true, and the initial segments are not identical.

Similar considerations can be made with regard to casual necessity, also as argued earlier.

Let's consider now two potential objections:

Objection 4: The difference between S and S' described above is not intrinsic, whereas the difference between T(0) in scenario 2 and S in scenario 1 or scenario 3 is intrinsic.

That objection claims that there is a difference between T(0) and S, but does not explain how there is that difference, or what kind of difference that would be, or why it would be intrinsic. Saying that S is intrinsically like S' does not seem to address the problem, because by the descriptions, S seems indistinguishable from T(0), but relevantly different from S' - even assuming S' is coherent, which is not clear, either (e.g., how can we make sense of a person who does not change at all, not even in his states of mind? It seems he wouldn't even think, feel, or do anything, not even in his mind; if so, how can he be conscious?).

Objection 5: God's state of mind at S resembles more God's state of mind at S' than it resembles the state of mind that God would have at an initial temporal state, like the state T(0) presented in scenario 2.

Leaving aside that it's not at all clear that it's coherent to posit any state of mind – or any person, for that matter – in a purportedly timeless changeless state like S', objection 5 makes a claim of a difference without any explanation as to how the state of mind of God in S and T(0) would be different, and further, different in a sense that is connected to the issue of time.

Given that S and the first state at T(0) in scenario 2 appear not to be different at all, based on the descriptions – as argued above -, there appears to be no good reason to accept this objection, either, at least until a defender of a KCA+ can explain that difference in a coherent manner.

2.1.5 Metrically amorphous time.

While Craig rejects Swinburne and Padgett's account of undifferentiated time before creation [10], he contends that a variant of it would be compatible with God's creation.

So, someone might try to combine the KCA with an account like that, instead of timelessness. However. the account in question is obscure as well, and those defending it would have the burden of explaining what they mean. But moreover, for that matter, if the account is coherent, they would have to explain why God would be the only or most plausible candidate for existing in this kind of amorphous time, rather than some weird quantum thing, or space, or something along those lines.

2.2. Conclusions based on the analysis of the proposed creator.

Based on the analysis so far, in my assessment one ought to reject the KCA+ as defended by Craig, or relevantly similar versions, since:

a. It's not clear that the idea of a timeless agent is coherent.

b. Even if a timeless agent is coherent, it's not clear that the idea of an agent that is timeless sans creation but temporal with creation is coherent.

c. There are strong reasons to think that the creator proposed by Craig actually would be an entity that begins to exist but has no cause of his existence, contradicting premise 1 of the KCA.

d. If the creator proposed by Craig for some reason can be properly called 'timeless' sans creation, but temporal with it – which seems very implausible given the previous subsections -, it seems that plausibly, the same would apply to the universe if it had a first state, which on its own would block the KCA, as argued earlier.

It might be suggested that, perhaps, some versions of the KCA+ defended by other philosophers fare better.

However, as long as they claim that there is a personal creator that is timeless sans creation and temporal with creation, or make any similar claims, it seems that the same problems will likely arise, and such claim seems to be a key part of the KCA+, at least in all usual variants.

In any case, any defender of the KCA+ ought to explain what they mean when they say that the creator is timeless without creation and temporal with it, etc., and in particular, explain how it's not the case that God begins to exist in their model of creation.

All that said, I will address the premises of the KCA and the arguments offered in support of them in the rest of the essay.

3. The first premise of the KCA.

The first premise of the KCA states:

P1. Everything that begins to exist has a cause.

In this section, I will assess first the meaning of the first premise, and then arguments given in support of it.

3.1. Beginning to exist and coming into existence. An alternative principle.

In his defense of the KCA, William Lane Craig assumes an understanding of 'begins to exist'[11] which I will call 'C-begins-to-exist', and which can be stated as follows:

An entity X C-begins to exist at t if and only if the following conditions obtain:

CBTE-i. X exists at t, and it's a tensed fact that X exists at t.

CBTE-ii. There are no states of the world at which X exists timelessly.

CBTE-iii. Either there is no t' < t such that X exists at t', or any t' < t at which X existed is separated from t by an interval of positive duration.

In that definition, X ranges over entities, and t over instants or intervals of finite, non-zero duration.

Let's briefly analyze the conditions:

CBTE-i. X exists at t, and it's a tensed fact that X exists at t.

The requirement that X's existence at t be a tensed fact, in particular, rules out beginning to exist if a tenseless theory of time is true. It seems odd to me that Craig would include tense in the definition of "begins to exist", but he argues that, under a tenseless theory of time, a universe with a first event did not begin to exist just as a meter stick does not begin to exist just because it has a first centimeter.[11]

That argument sounds odd to me as well, since while a meter stick does not begin to exist in virtue of having a first centimeter, that's not relevant, since having a first centimeter is a spatial, not a temporal claim, while 'begins to exist' - at least, in this context – is clearly a claim about time, not space.

In fact, the stick in question does have a beginning in space because it has a first centimeter - at least if there is no demand for an arbitrarily precise spatial beginning -, and similarly, even if a tenseless theory of time is true, the stick does have a beginning in time if there is, for instance, a first year at which it exists – at least if there is no demand for an arbitrarily precise temporal beginning, and a fuzzy beginning will do.

It is true that in order to say that the stick has a spatial beginning or a first centimeter, we need to pick a direction in space to say which centimeter is first – explicitly or by context -, whereas in the case of time, the direction from past to future is already implicit in our language.

But that does not appear to be a difficulty for a temporal beginning of existence under a timeless theory of time. Intuitively, to say that the stick begins to exist seems to be the same as to say that the stick has a temporal beginning, or a beginning in time, and then it follows that the stick does begin to exist, regardless of whether a tenseless theory of time is true.

CBTE-ii. There are no states of the world at which X exists timelessly.

This is an odd condition since the term 'timeless' is odd on its own. It's not clear to me that it's being used coherently by Craig. But I've already addressed the matter of timelessness, so I will go no further on this point, and assume from now on and for the sake of the argument that the concept of C-begins-to-exist is a coherent concept.

CBTE-iii. Either there is no t' < t such that X exists at t', or any t' < t at which X existed is separated from t by an interval.

This condition, together with the fact that t ranges over instants or intervals of positive finite duration, entails that if there is a finite open interval (0,t] at the beginning of the universe, then the universe began to exist. It's a debatable matter given that the metric might not be intrinsic, but I will no go further on this, and I grant this point for the sake of the argument.

In addition to giving that definition, Craig understands 'begins to exist' and 'comes into being' as semantically identical. In fact, he actually defines 'begins to exist' to mean the same as 'coming into being', and then defines "comes into being". [11]

I don't think that that captures the meaning of 'comes into being' or 'comes into existence', and in any case, I think it's relevant to consider some alternative principles in order to assess the arguments in support of the first premise, and one of them involve objects coming into existence in a different fashion, so I will not follow Craig on the usage of 'comes into being'.

The idea of coming into existence seems to imply a change in the state of the world, plausibly from a state at some time t' at which some entity X does not exist, to a later state at a time t > t', at which X does exist.

But perhaps, someone might suggest that the idea of coming into existence does not require that the previous state of the world at which X does not exist be temporal, and a causally prior state suffices. I'm not sure how a state might be causally but not temporally prior than another state – at the very least, it's very counterintuitive to me-, but I will leave that open, and define 'comes into existence' as follows:

An entity X comes into existence or comes into being at t if and only if the following conditions obtain:

CIE-i: X exists at t.

CIE-ii: Either there is a time t' < t such that X does not exist at t', or in any event there is a state of the world S at which X does not exist, and S is causally prior to the state of the world at time t at which X exists.

CIE-iii: Either t is the first time at which X exists, or t is separated from any previous time at which X exists by a temporal interval of positive duration.

In this context, X is also any entity, and t any instant or any temporal interval of positive duration.

Given those definitions, let's consider another potential principles, as an alternative to premise 1.

P1': Everything that comes int existence has a cause of its existence.

I do not claim that the principle is true; it may well be challenged.

I offer it in order to compare it with the principle offered in premise 1 of the KCA, since I find P1' to be more intuitively plausible, at least if one assumes that there might be a beginning of time – a matter I will consider in the next subsection.

3.2. Arguments in support of the first premise.

In this section, I will consider arguments in support of the first premise given by William Lane Craig in his defenses of the Kalam Cosmological Argument [1 2 ], as well as other potential basis of support from premise 1.

3.2.1. Nothing comes from nothing.

Craig maintains that the principle that nothing comes from nothing is intuitive and supports the first premise. In this context, he points out that objects don't just pop into existence without a cause.

However, the examples he provides as obviously counterintuitive are all examples of objects coming into existence without a cause, in the sense in which I defined 'coming into existence' earlier.

It is true that those are also examples of objects that would in those scenarios begin to exist without a cause, and even C-begin-to-exist without a cause if a tensed theory of time is true.

However, it might be that the feature that makes it intuitive that those objects have causes is that they come into existence, not that they begin to exist. Or maybe it's some other feature.

On that note, there are at least some reasons to suspect that that our intuitions – or at least mine – support the conclusion that it's more likely to be the fact that they come into existence that makes it intuitive that they have a cause, rather than the fact that they begin to exist, or that they C-begins-to-exist, especially if one considers a scenario in which time itself begins.

In particular – at least, after considering the matter -, I find the idea that time begins to exist counterintuitive, but if we consider a scenario in which time begins to exist (or even C-begins-to-exist), I do not find it intuitive that it comes into existence, or has a cause, or that all objects that exist at the beginning of time come into existence, or have causes. On the contrary, it seems very counterintuitive to me that time would come into existence. The same applies to, say, spacetime, or the universe if 'universe' is used in a way such that the universe begins to exist with time.

So, that would support premise 1' as intuitive[13], but not so much premise 1 if we factor in potential scenarios on which time begins.

3.2.2. Another alternative principle.

After further considering my intuitions on the matter by contemplating more scenarios (but which don't involve a beginning of time), it also seems plausibly intuitive to me that for every object O that exists at some time t, there is some time u < t, and some object U at u which is a cause of O's existence at t.

If that is so, then perhaps it's neither the fact that an object E comes into existence at t, nor the fact that E begins to exist at t, what makes it intuitive that there is a cause of E's existence.

Rather, perhaps it's the fact that E exists at t which makes it intuitive that there is an object E' at some earlier time t' which is a cause of E's existence at t, and thus there are causes of E's existence at any time.

If so, then our intuitions support P1':

P1'': For every object O that exists at t, there exists a time u < t, and some object U that exists at u, such that U at u is a cause of O's existence at t.

Here, t and u might be instants or finite intervals, and U might or might not be O at a previous time.

P1'' does seem intuitively plausible to me. But P1'' would be a problem for the defender of the KCA, since P1'' is in conflict with the second premise of the KCA, since it leads to an infinite past.

At this point, a defender of the KCA might argue as follows:

If out intuitions support P1'', then they also support the weaker P1, and we should then accept the first premise based on that. That the intuitions supporting P1'' would be in conflict with premise 2 is not a problem, since independent arguments can be given in support of premise 2, which would then defeat P1'', but not P1.

However, once one considers that kind of scenario, then premise 1 appears counterintuitive, at least to me.

In fact, once we stipulate that there is a finite past in order to contemplate that scenario, and the universe (or space, spacetime, etc.) begins to exist at some initial time, then my intuition is that it does not come into existence, and it does not have a cause.

Furthermore, the hypothesis of a timeless cause of time is difficult to make sense of even if the hypothesis does not posit a cause that changes, and plausibly incoherent if it does. In any event, even if it does not change, the claim is highly counterintuitive to me – definitely more so than something that begins to exist but does not come into existence, and has no cause.

So, taking the hypothesis that time begins to exist as a live option – which seems reasonable in principle, even if it's not intuitive to me at this point -, that takes away support for P1'', but also for premise 1.

On the other hand, premise P1' remains unaffected, so I will continue comparing premise 1 with the alternative premise 1'.

3.2.3. Are universes an unjustified exception?

Another argument that Craig makes is that making an exception for the universe would be ad-hoc. He asks why wouldn't ordinary objects come into existence all around us. Leaving aside for now the vagueness of the word 'universe', there are differences that, intuitively, are relevant. In particular:

1. Assuming that the universe had a beginning and that that is the beginning of time itself, then there is no time prior to the first state of the universe. So, if P1'' is intuitive – as it seems to be -, then a universe with such a beginning would already be an exception to that intuitive principle.

On the other hand, in the case of the objects we see around us, there earlier times, and earlier causes. Intuitively, that seems to make relevant difference.

2. Another potentially intuitively relevant difference would be that those objects around us that begin to exist also come into existence, whereas in case time had a beginning, any object that existed at the beginning of time would only come into existence if there is a state of the universe that is causally prior to time. That is at least more counterintuitive than a universe that begins to exist but does not come into existence.

At this point, someone might object to that and suggest that my intuitions are unusual. [13]

In particular, didn't scientists continue to look for a cause of the Big Bang? Why didn't they stop looking for causes? Are they intuitions different from mine?

However, the fact is that the Big Bang model does not provide an understanding of the universe beyond a certain point in the past, where effects from forces other than gravity should be taken into consideration.

In other words, scientists are trying to figure out the causes of a very hot, dense, and small universe that existed about 13.7 billion years: indeed, we don't know the causes; a theory that only considers gravity but no other forces is inadequate to provide a full understanding of it.

But those scientists seem to be asking the question: 'Where caused the state of the universe in which it was dense, small and hot?' (or similar ones), on the understanding that before the first state of the universe that can be reliably described with present-day models, there were other states of the universe that are beyond the descriptive capabilities of current scientific understanding.

Moreover, while it may be that if some scientists come to the conclusion that our universe began to exist, in some limited sense of 'universe', then they would not assume that time did, and so they might look for causes earlier in time – for instance -, or generally that there were no earlier universes in that limited sense of universe, and which would be part of the universe in the sense in which the word 'universe' is used in the second premise of the KCA. [14]

Also, and additionally, modern physics is in many cases counterintuitive, and so scientists may well look for counterintuitive causes as well, if their models – which make successful predictions in most or nearly all situations – suggest that they probably exist. However, in this context, I'm analyzing the intuitiveness of the premise, and so that would not affect my assessment.

In the end, readers will make their own assessments of course, but I don't think my intuitive assessments are unusual. Also, even though my intuitive assessment of infinite past vs. a finite past has changed over the years – while I used to find both of the alternatives counterintuitive (so, surely, one of my intuitions was wrong), I used to find the former more counterintuitive, whereas now it's the latter –, my intuitions that assuming that time and space began to exist together in a finite past, there probably is no timeless cause of time or space – which would be even much less intuitive than space and time having no cause -, have not changed, so if anything, those intuitions seem firmer.

That said, there are difficulties with appeals to our intuitions about time and space in contexts such as the purported beginning of time, such as:

a. There seem to be significant differences between the intuitions of different people. There might even be differences between their pretheoretical intuitions. So, plausibly, and at least on some the issues under discussion, there might not be one single human normal pretheoretical intuition on a matter.

b. The intuitions of a person on some of these matters may well change over time, as that person learns more about physics, cosmology, philosophy, etc., and sometimes it's difficult to figure out what a person's pretheoretical intuitions are.

c. There seems to be no particularly good reason to think that pretheoretical intuitions are more reliable than intuitions developed later – let alone more than the intuitions of specialists, like cosmologists.

d. In any case, modern physics shows that in unfamiliar environments like something very small or massive, things are 'weird', and our intuitions about time, space and related matters do not seem to work well. But the universe a long time ago was both very small and massive.

Of course, none of that represents a problem for use of our intuitions in daily life or relevantly similar cases, but it raises questions about their reliability in cases like those involved in the context of discussions of the KCA.

In addition to the question of which principle or principles are intuitive, another question is whether empirical evidence supports the first premise. I will address the matter in the next subsection.

3.2.4. Empirical evidence.

Craig also claims that empirical evidence supports premise 1, and points to a vast number of examples. However, there that argument does not seem to work, since:

a. The examples that Craig give would also support P1'', which would contradict the second premise of the KCA.

b. If one contemplates a scenario in which time begins, and one also grants that premise 1' - which is intuitively plausible -, then the prior probability of premise 1 under that scenario is very low – at least by my intuitions -, and the observations offered by Craig as empirical evidence in support of premise 1 are not more likely if premise 1 is true than they are if premise 1 is false given premise 1', so they do not raise the probability of premise 1, either.

In addition to that, using a similar argument from empirical evidence, we may find support to propositions such as:

EE1: Every intelligent being has at least one non-intelligent cause.

EE2: Every agents has at least one cause that is not an agent.

EE3: Every personal beings has at least a non-personal cause.

And so on. Granted, a defender of the KCA + might give other arguments against those inferences, but that's a matter they would have to deal with when defending the proposed cause of the universe, even if they managed to establish that the premises of the KCA are correct, and even leaving aside the problems with the concept of the agent that they propose.

4. The Second Premise of the KCA.

The second premise of the KCA states:

P2. The universe began to exist.

In addition to the question of the meaning of 'begins to exist', an important issue is the meaning of 'universe'.

According to Craig, in the second premise of the KCA, the universe is the whole of material reality. [14].

That does not seem entirely clear to me, since 'material' is not so clear, but it is important to note that if 'universe' is used in a limited sense so that a universe can be caused by early universes, events in a multiverse, etc., that limited usage may be of interest in scientific cosmology, but it's a different usage.

So, in this context, it seems to me that premise 2 should be understood as entailing that time began to exist, though someone might say that it would be enough for a defender of the KCA to show that space and all that is 'material' began to exist, even if the arguments proposed in defense of the KCA fail to rule out an earlier time when – for instance – there were only unembodied agents.

Either way, it is apparent that arguments showing that the universe, in a limited sense of 'universe' begin to exist, would fail to provide support for the second premise of the KCA.

In the rest of this section, I will assess several arguments given in support of premise 2, defended by Craig, Koons, and Pruss, and I will argue that they all fail to support the premise in question.

4.1. Hilbert's Hotel

The first argument I will consider is the 'Hilbert Hotel' argument, intended to show that an actual infinity is metaphysically impossible. Based on that, Craig argues [15] that an infinite past is metaphysically impossible.[16]

According to Craig, the 'Hilbert Hotel' argument shows that an actual infinity is counterintuitive, and based on that, he claims that an actual infinity is plausibly metaphysically impossible.

The Hilbert Hotel is a hotel with a denumerable number of rooms, [17] and Craig maintains that it's the actual application of the concept of infinity to the real world, rather than the consideration of abstract sets, what brings the counterintuitiveness of an infinity to one's attention.

Of course, a hotel like the one he proposes is counterintuitive, since (for example) we would never be able to build it, we wouldn't be able to communicate with the rooms in real time as in the proposed scenario, etc. However, all of that particular counterintuitiveness seems to have everything to do with an infinite hotel, and nothing to do with an actual infinity.

On the other hand, the alleged counterintuitiveness claimed by Craig seems to result from a misunderstanding of the meaning of the words, and disappears once the claims are clarified.

To see this, let's consider some of the scenarios he proposes as supporting the claim of counterintuitiveness:

First, Craig presents a scenario in which all of the rooms are occupied, and a new guest arrives.

According to him, somehow the fact that there are no more guests after the arrival is a problem, or counterintuitive. However, that is not the case, once it's clear what one means by 'more guests', for the following reasons:

In the sense of cardinality, there are no more guests after the new guest arrived than there were before. However, that merely means that there is a bijection between the set of guests before the new arrival, and the set of guests after the arrival, which is as clear as the fact that there is a bijection between the set of positive integers and the set of positive integers plus zero.

On the other hand, clearly there is at least one sense in which there are more guests after the new arrival than there were before, namely the sense that all of the previous guests are still there, and there is also a guest who wasn't there before. In terms of sets, the set of guests before the new guest arrives is strictly contained in the set of guests after the new guest arrives. All of this is unproblematic.

Of course, shifting infinitely many guests from one room to the next would not be doable for any human being, alien from another planet, etc., but that's not relevant, since if any of that is a problem also for the metaphysical possibility of that hotel, it's clearly a problem for Craig's hotel, not for actual infinities in general, as is clear from the fact that such problems do not affect – for instance – the hypothesis that there are infinitely many galaxies, which is a hypothesis considered seriously in present-day science.

In addition to the previous scenario, Craig considers a scenario in which infinitely new guests arrive at the desk, and the proprietor talks to them, etc.

However, it may be that the meaning of the word 'desk' is such that it's not coherent to say that infinitely many show up at the desk. But that scenario in which infinitely many guests arrive is not a condition included in the definition of the Hotel in question, so if it is an incoherent scenario due to the meaning of 'desk', that is not enough to rule out the possibility of the Hotel.

On the other hand, if there is nothing in the meaning of the word 'desk' preventing that infinite arrival, then still things like building such a hotel, or communicating at arbitrarily fast speed with infinitely many people, etc., look counterintuitive.

However, as before, any counterintuitiveness is not a consequence of assuming an actual infinity, but of assuming the infinite hotel that Craig and Sinclair describe. For instance, also as before, there seems to be no similar counterintuitiveness in the hypothesis that, say, there are infinitely many galaxies.

As for the number of guests after denumerably many new ones arrive, it's the same as before the arrival in terms of cardinality, but again that merely means that there is a bijection between the set of guests before the arrival and the set of guests after the arrival, which is as clear as the fact that there is a bijection between the set of positive integers and, say, the set of even positive integers.

On the other hand, there is at least one sense in which there are infinitely more guests after the arrival, namely, that all of the guests who were in the hotel before infinitely new ones arrived remain there, and infinitely many guests who were not in the hotel before the arrival are in the hotel after the arrival.

That is also unproblematic. As long as one keeps in mind what's meant by 'more guests', any counterintuitiveness related to the 'more guests' issue is gone, and any counterintuitiveness left has to do with the specific features of an infinite hotel.

Similarly, if one guest departs, there are as many guests as before in the sense of cardinality, but there is a sense in which there were more guests before the departure, namely that all of the guests that are in the hotel after the departure were in the hotel before the departure, but there was also one guest before the departure that is no longer in the hotel after she departed (obviously). In terms of sets, the set of guests after the departure is strictly contained in the set of guests before the departure. Once again, all of this is unproblematic, and any issues related to the number of guests do not appear to be counterintuitive as long as one keeps in mind what one means by 'more guests'. Now, if denumerably many guests leave, whether there are as many as before in the sense of cardinality depends on which set we remove, just as we can subtract an infinite set A from the set of natural numbers N such that N\A has the same cardinality as N (e.g., A:={x: x is a prime natural number}), or subtract a set B such that N\B is finite (e.g., B:={x: x is a natural number greater than 1}).

As before, there are no problems in terms of how many guests there are, or in terms of subtractions.

Craig raises other scenarios based on Hilbert's Hotel, but they're handled in a similar fashion, namely once it's clear what one means by 'more guests', etc., it should also be clear that any of the issues raised by Craig vanishes, and any counterintuitiveness is not the result of the fact that the scenario involves an actual infinity, but of the fact that it involves an infinite hotel, with certain specific properties, like infinitely people arriving at once, a proprietor communicating with and moving infinitely many guests, etc.

So, in short, the 'Hilbert Hotel' argument is not a good argument against the possibility of actual infinities.

The Hotel itself appears counterintuitive, though it's not clear at all that it's metaphysically impossible. In any event, even if Hilbert's Hotel is impossible because of particular features of the Hotel, that would not give us any good reason to think that all other infinities are impossible, like infinitely many galaxies, or – more relevantly in this context – infinitely many past events, one before the other, and Hilbert's Hotel provides no good reason to suspect that those other scenarios aren't possible.

4.2. Grim reapers.

The 'grim reapers' argument has different forms, and it may be offered in support of the hypothesis that actual infinities are impossible, and/or that infinite temporal regress is impossible, or that time is necessarily discrete, among others.

In this subsection, I will focus on some of Alexander Pruss' arguments. [18]

Briefly, the scenario is as follows:

Fred is alive at t₀– 11.00 am. in Pruss' scenario -, and then, there is a grim reaper (say, GR_n), set to kill Fred at t_n=(t₀+(1/n) seconds), if Fred is alive. Fred cannot survive a grim reaper attack, and nothing else possibly kills him.

At first glance, there are some difficulties that might affect our intuitions and assessments, such as:

a. Death is a gradual process, and also the words that we use to describe the world around us are imprecise.

b. Modern science still does not have a way of handling arbitrarily short amounts of time, and our regular intuitions about time and space do not seem to work well when certain quantum effects are involved, etc.

But let's say that difficulties of that sort are resolved. Then, there is another potential difficulty:

Pruss' 'grim reapers' scenario is strictly logically impossible, but an infinite regress of events is not.

In other words, assuming a scenario in which there is an infinite regress of events, a contradiction does not follow. Instead, what would be impossible in that scenario, would be Pruss' construction.

I suppose there might be a way around that potential difficulty, by rewording the argument as an argument forcing us to choose between different intuitions, though, so let's leave that aside for now as well.

4.2.1. Discrete time.

Pruss distinguishes between the following two types of discrete time.

1. Time is rigidly discrete if there necessarily is a minimum temporal unit.

2. Time is discrete in an Aristotelian sense if there are in fact finitely many moments of time, but each interval can be subdivided infinitely many times.

In his version of the grim reapers argument, Pruss is assuming that time is not rigidly discrete. He argues against rigid discreteness based on relativity, and on Zeno's Stadium.

I don't find Zeno's Stadium to be a persuasive argument against rigid discreteness, but I won't address the matter here. I will just point out that the Stadium is at least a contentious matter.

As for relativity arguments, they seem to be based on the usual interpretation of time in physics, which holds that time is relative, and is based on empirical evidence. That would not be a problem on its own, in my assessment, since I don't see any good reasons to think that our observations are illusory and metaphysical time is actually absolute.

However, temporal relativity seems to be problematic for theism. Let's consider the following scenario:

There are astronauts on Mars, and one of them prays to the creator, asking her to establish a communication between her (i.e., the astronaut) and her loved ones on Earth, with no more than a, say, five seconds delay (or five minutes, for that matter).

Then, it seems to me that even if there were a creator of the universe and she were listening, she would not have the power to grant the request even if she wanted to, since that would seem to result in paradoxes, resulting from sending information faster than the speed of light.

Granted, someone might suggest that, perhaps, metaphysical time is relative but in a way very different from what modern physics proposes, but if one leaves aside modern physics, it seems one also leaves aside any support for the hypothesis that time is relative.

In particular, without the speed of light limitation, there would be no support for the hypothesis that time is relative, since we would be leaving aside a crucial part of the empirical basis for the hypothesis that time is relative.

Also, someone might suggest that physical time is relative, but not metaphysical time. But what matters in this context is metaphysical time, and as long as it's not relative, it seems relativity provides no evidence against rigidly discrete time.

But potential consequences of accepting relativity for theism aside – which, as a non-theist, I would not find problematic -, let's grant that time is not rigidly discrete for the moment, and let's consider Aristotelian discreteness.

The definition of Aristotelian discreteness given by Pruss might give the impression that it implies immediately that there is no infinite temporal regress of any kind – since it's stipulated that there are only finitely many moments -, but that would not be a proper interpretation of Aristotelian discreteness, which is clear by the fact that he argues then against infinite regress under Aristotelian-discrete time rather than just point out that it's immediately implied by the definition, and even more clearly by his proposed specification of Aristotelian discreteness. [18]

To simplify, we may consider the following alternative definition.

3. Time is discrete in the Aristotelian-2 sense if and only if, necessarily, there are finitely many moments [19 ]of time between two given ones, but any temporal interval is possibly subdivided in smaller subintervals.

Alternatively, we may consider Pruss' suggestion. [18]

I will use the definition above, but the reply I offer works equally under Pruss' suggestion.

So, Aristotelian-2 discreteness is clearly incompatible with an infinite regress of events plus a time at which Fred is alive before that infinite regress.

Thus, if there were an infinite regress of events under Aristotelian-2 discreteness, that would be an infinite sequence of events without any time prior to the whole sequence, and so Pruss' argument from grim reapers would not apply to it, since Fred could not be alive before the sequence.

Hence, Pruss' argument is not applicable if time is discrete in the Aristotelian sense.

On the other hand, Pruss argument is also not applicable if time is rigidly discrete. [18]

So, it seems that if the argument under consideration were successful, it would work as an argument against the hypothesis that time is possibly dense. Different arguments would still be required to rule out an infinite past if time is necessarily. I will consider one argument given by Pruss under Aristotelian-discrete time later.

4.2.2. Grim reapers, infinite regress, and the Hilbert Hotel.

According to Pruss, if an infinite temporal regress were possible, then the Hilbert Hotel would be possible. His argument from infinite temporal regress to the Hilbert Hotel holds that assuming that an infinite series of events going backwards in time is possible, each event could be the creation of one room, and no room gets destroyed.

Also, he argues that if Hilbert's Hotel were possible, it would be possible for there to be one reaper produced in each room, and it would be possible for the Hotel's staff to set the reaper in room #n to kill at t_n.

So, Pruss claims that if the Hilbert Hotel were possible, then some other things would be possible, and a contradiction would follow.

Yet, the Hilbert Hotel, on its own, appears not to be contradictory. If so, clearly it does not entail a contradiction. So, Pruss appears to be using the Hilbert Hotel plus some criterion or criteria for metaphysical possibility, in order to derive a contradiction. Yet, Pruss does not say what criteria he's using, or why he considers that criteria reliable, which obscures the matter considerably.

Still, apparently, Pruss is not using the a criterion of conceivability without visualization, because that kind of criterion would seem to support the possibility of the Hilbert Hotel, but obviously not the possibility of a contradiction.

Also, he's apparently not using only a criterion that requires visualization, due to the infinities involved. Moreover, it seems that in order to establish the possibilities of each grim reaper, he's not even using a visualization criterion combined with other criteria, since he says that the grim reapers need not be physical entities, but might be non-physical entities with the specified causal powers, and even though the meaning of 'non-physical' is unclear, given usual examples, it seems plausible that they might not be visible.

So, even ruling out a criterion of conceivability without visualization as a guide to metaphysical possibility for the sake of the argument, given that Pruss does not specify which criterion or criteria he is using, we may point out the following:

1. It's not clear that if an infinite temporal regress were possible, it would be possible to make rooms like that. In particular, if metaphysical possibility reduced to causal possibility – as Pruss suggests elsewhere[20 ], though I'm not convinced by that argument -, there would seem to be no good reason to suspect that just because an infinite temporal regress is possible, there will be some entity with the power to bring about those rooms.

2. Even if the rooms were brought about, it's not clear that they would last for long enough for there to be infinitely many of them at any time.

3. Even if infinitely many rooms at once were possible, it's not clear that that would be a hotel. For instance, it might be possible that infinitely many rooms exist, yet the speed of light limits communication between them. In particular, for every room, only finitely many ones would be within its light cone. In that case, there would be infinitely many rooms, but they would not constitute a hotel, or anything that would work as required in this argument.

4. As suggested above, it's not clear that the rooms and/or the staff and/or anyone else would have the power to make grim reapers with the power to check whether Fred is alive from an arbitrarily large distance, and kill him instantly, or almost instantly.

5. Also, it's not clear that the staff or anyone else would have the power to make devices that properly measure arbitrarily short temporal intervals, etc.

6. In particular, it's not clear that all of those things would be possible together if an infinite temporal regress were possible.

Perhaps, Pruss is using a criterion or a set of criteria for assessing metaphysical possibility that is broad enough to allow him to establish that all of that would be possible if an infinite regress of events were possible. But the questions remain: which criterion or criteria is he using, and why should we accept them?

In any case, given the broadness of whatever criterion or criteria Pruss is using, we may reason as follows:

It seems clear that an infinite forward sequence of events {E(n)} is possible. That would not even require an actual infinity, but merely a potential one.

So, let's consider the following scenario – which we may call the 'Boundless Innocent Suffering' (BIS) scenario:

Let's start with some fixed year, which we may number '1'.

For n > 2, the event E(n) is as follows:

a. E(n) begins at the beginning of the year n!, and ends at the beginning of the year 1+2*n!

b. In the first year of E(n), 1 torture chamber with 1 being with a mind similar to that of a four-years-old normal human, and equal capacity for suffering in it (say, Bob(n)) is produced. Producing it does not need to take the whole year. It might take a second, a minute, a month, etc., but in any case, by the end of the first year, the torture chamber with Bob(n) in it is fully functional.

c. Since the torture chamber with Bob(n) in it is ready, and until the end of E(n), Bob(n) is burned alive from the waist down, allowed to regenerate – which takes a few seconds -, then burned alive again, and so on. So, this torture lasts for at least n! years.

So, for instance, Bob(3) is horribly tortured for at least 6 years, from the year 7 to the year 13, Bob (10^{1000000!!!!!!!}) is horribly tortured for at least ((10^{1000000!!!!!!!})!)years, from the year (1+((10^{1000000!!!!!!!})!)) to the year (1+2*((10^{1000000!!!!!!!})!)), and so on.

Granted, the previous scenario would only a problem for claims of necessary existence of a sufficiently powerful and morally perfect creator, or similar claims, and not necessarily for the KCA. However, the point is that a sufficiently broad criteria for metaphysical possibility appears to create significant problems for usual versions of theism, and I think that's a matter to take into consideration when assessing arguments usually intended as part of a case in support of such theistic views.

All that said, I have to admit that the amount of suffering that actually exists in the world is much more than enough for me to conclude that an evidential argument from suffering is successful.

Still, it seems to me that the BIS scenario may well be too much even for most of those who reject arguments from suffering in the case of the real world, in particular because in the BIS scenario, there is no silver lining. BIS never ends, and the torture keeps getting longer and longer.

So, while it's true that the suffering of each victim might end (that's not specified in the BIS scenario, so the torture might or might not continue after the stipulated period), for any natural number n and for any time t, there are more than n victims still to come in the future, each of which will be tortured for more than n years, and so on.

Leaving aside potential difficulties the criterion or criteria used by Pruss as a guide to metaphysical possibility and/or for theism, one may still point out that even if the Hotel with the reapers, mechanisms, etc., are possible, that would not entail that Fred cannot die unless he's killed by a reaper. Perhaps, for all we know, Fred's life extends through an interval [u, t₀] or (u, t₀] for some u prior to t₀, and the reapers would have no one to kill, resulting in no contradiction.

On that note, Pruss suggests[18] stipulating that there is nothing in the world that could kill Fred except for an individual grim reaper. Apparently, under that stipulation, Fred couldn't kill himself of die on his own of natural causes, either.

However, there seems to be no good reason to suspect that if an infinite temporal regress is possible, then it's possible that nothing but a grim reaper can possibly kill Fred in that scenario.

For that matter, we may suggest the following:

i. If a finite past is possible, and then it would be possible that nothing but a grim reaper can possibly kill Fred. Then, if a finite past is possible, usual (usual in philosophy) forms of monotheism are false.

But since whether a finite past is possible has nothing to do with which entities could possibly kill Fred, for that matter we may suggest, in addition to i.

ii. If an infinite past is possible, then it would be possible that nothing but a grim reaper can possibly kill Fred. Then, if an infinite past is possible, usual forms of monotheism would be false.

From that, we may conclude that usual forms of monotheism are false.

Of course, that wouldn't be a good argument against forms of monotheism usual in philosophy – or any others -, since we would just be stipulating for no good reason whatsoever that nothing could possibly kill Fred in those cases. But Pruss' argument has the same problem – namely, the suggested stipulation would be unwarranted, and thus this 'grim reapers' argument would fail even if we granted all of the previous claims about the possibility of the hotel, the grim reapers, etc.

Granted, someone might suggest adding to Pruss' scenario not the stipulation that nothing else could kill Fred, but simply that nothing else does kill Fred. However that would still seem to be a case of constructing by stipulation a scenario that is contradictory, and conclude from there that that rules out the possibility of scenarios that are not contradictory, namely all of the scenarios in which there is infinite regress, time is not rigidly discrete, and by construction the scenarios do not entail a contradiction. Without explaining what principles are used as guides to metaphysical possibility, it seems to me that the argument has not been sufficiently supported.

In any event, the burden seems to be on the defender of the 'grim reapers' argument, and it's not been met yet, to the best of my knowledge.

4.3. Grim placers and/or grim signalers.

In this subsection, I will address a variant of the argument from grim reapers, defended by Robert Koons[21], in which each grim reaper, or rather grim placer, checks whether there is some Fred particle at a specified location, and if there isn't, he places one at some specific location. Otherwise, he keeps the particle where it is.

According to Koons, his argument shows that time is not dense, and that there is no infinite past.

I disagree with that assessment, and furthermore, I will argue that some of his assumptions may be used, in combination with no less plausible ones, to make an argument against usual forms of philosophical monotheism.

4.3.1. Possibility of a grim placer.

Among other hypotheses, the argument assumes the following:

1. There is a region R of duration d in a possible world W, and a grim placer G_d, such that GP_d has the intrinsic power and disposition to do as follows:

a. If there is no Fred particle at any distance y<d from a fixed plane P, then G_d creates and places a Fred particle at a designated location exactly d meters from P.

b. Otherwise, GP_d keeps any Fred particle that is closer to plane P in its position.

It's not clear to me what criterion Koons is using, as a guide to metaphysical possibility, in order to assert that one such scenario is possible.

It seems to me that Koons probably isn't using a conceivability without visualization criterion, because such a criterion would seem to lead to the conclusion that there is no concrete being that exists necessarily, contradicting Koons' stance on that matter.

Also, it seems to me that we can't visualize the scenario, either. For instance, a requirement of the scenario is that the location of the particle is at exactly p meters from a certain plane. But it seems to me that we can't visualize anything with that kind of precision.

So, perhaps, Koons is using some other kind of conceivability criterion that is not broad enough to be a problem for Koons' position, but broader than a criterion that requires visualization. It seems to me that the burden to explain and defend the criterion in question would be on those who make claims of metaphysical possibility based on it.

Still, in any case, intuitively, it seems to me that the hypothesis that one of the E(n) of BIS is no less plausibly possible than Koons' assumption above. So, I would be inclined to grant point 1. for the sake of the argument, but also assume that at least for some n₀, E(n₀) is possible.

Granted, it might still happen that Koons uses a criterion to assess metaphysical possibility that blocks all of the E(n), but allows assumption 1. However, it's hard to see what kind of criterion would have that effect, without being ad-hoc.

In any event, as before, I would say anyone using such a criterion would have the burden of stating it clearly and defending the claim that that criterion is a good guide to metaphysical possibility.

Still, in any case, and alternatively, we may grant point 1. for the sake of the argument, without even assuming that any of the E(n) is possible, but assuming instead that the following scenario – which we may call the 'horror for a minute (HAM) scenario – is possible:

An entity with a mind, capability for suffering, regeneration, etc., similar to those in the BIS scenario, suffers horrible, tremendous pain – like being burned alive from the waist down, without any decrease in pain – for one minute.

Granted, someone might still reject the metaphysical possibility of H A M, while claiming that point 1, obtains, but that also put an even heavier burden on the person making such a claim.

4.3.2. Compressibility of spacetime.

Another assumption of the argument – though this one is an assumption for a reductio – can be stated as follows:

2. If there is some object A with an intrinsic property Q in a region R of finite duration d in a possible world W, then:

a. There is a function f from the parts of R to the parts of some region R' of some possible world W', such that f is topology-preserving and compresses time and space by half. For instance, if the duration of R is d, the duration of R' is d/2

b. There is a counterpart A' of A and a counterpart Q' of Q, such that A' intrinsically has property Q' in R'.

Point 2. is used for a reductio against temporal density. Koons argues that 2. is reasonable under the assumption that time is dense.

If there is an intrinsic metric of time, it seems to me that we don't know whether some properties aren't compressible, whether some processes require at least some amount of time, etc., even if time is dense.

But let's say that that is not the case, and let's grant 2 for the sake of the argument.

Then, it seems to me that similarly, and regardless of whether time is dense, we may accept a principle of decompressibility of time, or DT, which is at least equally intuitive, and which we can state as follows:

2'. If there is some spatial and temporally bounded region R of duration d in a possible world W, then there is a duplicate R' of R in possible world W', which contains counterparts of every object in R with counterparts of its intrinsic properties, but expands time by a factor of two.

DT takes no stance on whether space is also expanded.

It may be argued that DT is in a sense stronger than Koons' principle, so a more tailored approach might be required for an atheistic argument, but if we assume that no properties are incompressible, no processes require a minimum amount of time, etc., and we accept 2 . for that reason, it seems to me that it's equally plausible that a bounded region is possible temporally expanded by a factor of two.

But if one accepts DT plus E(n 0 ) of BIS, then there are possible duplicates of E(n 0 ) of arbitrarily long duration, provided that the properties of the torture chambers and the beings in them are intrinsic to the scenarios, which we may stipulate as Koons stipulates that the properties of the reapers are intrinsic. Those duplicates wouldn't be exactly the other E(n), for n different from n₀, but the fact is that there would be arbitrarily long possible scenarios of those child-like beings getting tortur e d.

Alternatively, even without assuming that E(n 0 ) is possible, if the H A M scenario is possible, a relevantly similar result is achieved: namely, there would be arbitrarily long possible scenarios of those beings suffering horribly.

A potential objection to those scenarios of horrible extended suffering would be that, as time expands, so do the mental processes of the victims, which become slower, so they do not experience any increased suffering. However, given that the mind, capability for suffering, etc., of the victims is intrinsic to the individual scenarios – rather than, say, slower functioning minds -, it's not clear that this objection would succeed. It would depend on what counts as 'intrinsic', and that's unclear.

But let's say for the sake of the argument that all such minds would also become slower in that way, in the case of expansion of time, so that this objection to the scenarios of extended suffering described above is successful.

Even then, there are alternative ways of constructing similar scenario. For instance, one such alternative can be constructed if the following premises are accepted:

Pa. For n=1, there is a possible scenario like H A M, which lasts for a minute.

Pb. Plausibly, if for some k, there is a possible scenario like HAM that lasts for k minutes, then there is another possible scenario like HAM, but which lasts for k+1 minutes.

How plausible are a. and b.?

Intuitively, I would say at least no less plausible than Koons' premises. Pa. seems quite clear, and Pb only requires finite additions (i.e., there is no assumption of a scenario in which one such being suffers forever).

Granted, someone might either reject Pa., or accept Pa but hold that, perhaps, there is some number K, such that for every k≤K, possibly, an entity of the relevant kind undergoes the kind of suffering described in the scenarios for k minutes, but necessarily, no such entity undergoes such suffering for K+1 minutes. That seems intuitively odd.

Still, assuming that that kind of objection to the proposed scenarios of extended suffering succeeds and the conjunction of Pa and Pb ought not to be accepted, then it seems plausible to me that the same kind of objection may successfully be raised to the principles proposed by Koons, since they're no more intuitively plausible than the conjunction of Pa and Pb.

Yet still, also assuming counterintuitively and for the sake of the argument that while all objections to the principles proposed by Koons fail and Pa is true, but Pb is not and there is such necessarily maximum number of minutes K, then we can still construct some scenarios that are very problematic for theism, especially when combined with Koons' principles. For instance, one such scenario can be constructed as follows:

All of space and time can be covered by non-overlapping spatiotemporal regions of equal size that are spatially cubic and have a fixed temporal duration K.

So, let's consider a possible world W at which there is a spatially cubic spatio-temporal region C(W) of finite size with one of those beings suffering for K minutes. We may consider then similar regions in other worlds, but spatio-temporally moved so that the regions are not overlapping (which seems obviously possible), and then apply infinitary patchwork, covering all of space and time at some world W'. While none of those entities suffers for more than K minutes at W', the fact is that all of time and space is covered by that kind of torment. We may also stipulate that there are no other agents spatially located in any of the regions, so that the whole of space and time is basically covered with horrible suffering and nothing else.

4.3.3. Infinitary patchwork.

A key principle of Koons' argument is what he calls "infinitary patchwork" [21] (or IP), and which can be put as follows:

Let's suppose the following conditions obtain:

IP(1): W = {W_n} is a countable series of possible worlds, and R={R_n}, is a countable series of regions of those worlds, such that for all n, R_n is a region of W_n.

IP(2): f is a function from R into the class[22] of spatio-temporal regions of some world W, such that f preserves the metric and topological structure of each of the R_n, and such that, if n≠m, then f(R_n) does not overlap f(R_m).

Then, there is a possible world W', and an isomorphism g from the spatio-temporal regions of W to the spatio-temporal regions of W', such that the following obtains:

IP(C): The part of W_n within R_n is exactly like the part of W' within g(f(R_n).

Let's grant IP for the sake of the argument.

Then, and as long as the minds of entities with intrinsic mental properties do not become slower as suggested in the previous subsection, it seems that from the assumption that E(n 0 ) is possible, plus the DT principle, a scenario relevantly similar to BIS follows at once, since for the reasons given in the previous subsection, there would be worlds W(n) and regions R(n) of those worlds with arbitrarily long periods of such terrible torment, and then one can send each of them to a world W via a function that preserves the topological and metric properties of each of them, and then patch them together into another possible world W' via infinitary patchwork.

Alternatively, given infinitary patchwork plus DT and the H A M scenario, it follows that there is a possible world where beings similar to those in the BIS scenario suffer horrible pain – like being burned alive from the waist down – for finite but arbitrarily long periods, and where that situation goes on forever, so for every n, there will be more than n of those beings who will endure that suffering in the future, and for more than n years each.

Also, alternatively, if such minds would become slower, there are other scenarios that seem to work, as described in the previous subsection.

In my assessment, those conclusions would be a defeater for any usual form of philosophical monotheism – though it's true that in my assessment, the actual amount of suffering in the world is more than enough as a defeater.

Still, that's only a problem for usual variants of theism, but we still need to address the questions of whether Koons' argument shows that time isn't dense, that the past is finite, etc.

It seems that even all of Koons' assumptions, plus the assumptions he uses in order to attempt a reductio, still do not present any problem for temporal density, as the following scenarios show:

First, let's consider the following scenario S₁:

1. There is a denumerable sequence of temporal intervals {I_n=[t_n+1,t_n)}, for all natural numbers n.

2. Each interval I_n has length 10^-n

3. During interval I_n, or at least during the last portion of it, there is a Fred particle at the designated position 10^-nmeters from the designated plane P. There is no other Fred particle during interval I_n

That scenario does not contain any grim placers, but it's a scenario that one could get by applying infinitary patchwork to each of the intervals.

Second, let's add grim placers; now, the scenario S₂is:

1. There is a denumerable sequence of temporal intervals {I_n=[t_n+1,t_n)}, for all natural numbers n.

2. Each interval I_n has length 10^-n

4. During interval I_n, there is a grim placer #n, GP_n.

That's a scenario containing infinitely many grim placers.

Each grim placer GP_n places the particle at the designated position, or the particle is there for another reason, so the grim placer has no job to do.

Someone might suggest that there is a problem if the intervals are open in the direction of the past, and closed in the direction of the future.

However, that wouldn't seem to be a problem, either, as the following scenario S₃ shows:

1. There is a sequence of temporal intervals {I_n=(t_n+1,t_n]}, for all natural numbers n.

2. Each interval I_n has length 10^-n

4. During interval I_n, there is a grim placer #n, GP_n.

Someone might add another condition to the definition of grim placers, demanding that they must act instantaneously, and suggesting that that would be a problem for each of the individual scenarios.

However, in that case any problem in S 3wouldn't be related to the infinitary patchwork principle, but it would be a problem with each of the individual scenarios used for the patchwork. But then, that would not work as an argument against temporal density, which presupposes that the individual scenarios are all possible.

So, it seems that Koons' 'grim placers' argument does not work as an argument against temporal density, even leaving aside any atheistic parallel arguments. Similarly, they don't work as an argument against an infinite past, either, since similar counterexamples succeed.

4.3.4. Grim signalers.

Koons actually considers an objection like the one I just raised, and correctly points out that what is required is that some signals persist from one grim reaper, grim placer, or grim signaler, to the next.

So, he reformulates the argument without any particles, and replacing the grim placers with grim signalers. It's essential to a grim signaler to have the power of receiving a signal from a previous grim signaler, and to have the power to send a signal to a successor.

Koons considers a potential objection, based on the suggestion that the regions would overlap, and rejects it, positing adjoining non-overlapping intervals.

However, adjoining non-overlapping intervals aren't good enough. The power to send the signal to the next signaler is akin to the power to make the particle last into the next interval. That seems to defeat the argument, because the regions that are being patched would have to overlap, or the properties of the signalers would not be intrinsic despite Koons' claim, so the principle of infinitary patchwork would not be applicable.

On the other hand, the atheistic parallel arguments I gave above do not have that problem.

4.4. An infinity by successive addition?

According to Craig, on a tensed theory of time, the reality of temporal becoming makes it impossible for there to be an infinite past series of events of equal duration, since those changes would have to happen by successive addition, but by addition of one change or event at a time, it would never be possible to reach an infinity. [2 3 ]

While it seems clear to me that it would be impossible for there to be such an infinite series with a beginning point, a key question here is whether that is also the case if there is no beginning point.

There appears to be a significant difference between a case with a beginning point and one that does not have a beginning point, namely that in the case in which there is no beginning point, there are no two events and/or times separated by an infinite temporal distance, so the temporal distance to be traversed from any time to another is always finite.

As long as traversing finite distances is not a problem, it seems to me that that would block the argument. At this point, someone might still object to the infinite series of events without a beginning and ask how the infinite series of past events was formed in the first place, claiming that it would never form under a tensed theory of time.

However, a reply to that line of argument would be to point out that precisely, since there is no beginning point, there is no need for an infinite series to be formed from a certain time on, and also no time at which the infinite past wasn't already there.

Given the previous considerations, in my assessment, the fact that, on a tensed theory of time, temporal becoming is real, is insufficient to rule out that the hypothesis there is a beginningless infinite past, under the assumption that a tensed theory of time is true.

In the next subsections, I will arguments defended by Craig and which purportedly support his conclusion that it's impossible for there to be an infinite series of past events on a tensed theory of time, but before I address them, I would like to point out that, while Craig defends those arguments against an infinite past in the context of a tensed theory of time, my replies to those arguments do not require a tensed theory of time, though of course they do not reject it, either.

Thus, if someone defended arguments essentially like those defended by Craig but without assuming a tensed theory of time, the replies I give below would be equally applicable to them.

4.4.1. Tristram Shandy.

One of Craig's arguments against an infinite past is based on the story of Tristram Shandy. [23]

Shandy is a man who writes his autobiography, at a rather slow pace: it takes Shandy a year to write the events of a single day.

According to Craig, we should reject an infinite past because it's obviously coherent to write an autobiography at that pace, but if Shandy had been writing from infinity, that would lead to absurdities.

However, what is obviously coherent is to write such an autobiography starting at a specific day.

On the other hand, to write an autobiography counting from infinity is logically impossible.[24]

To see why this is impossible, let's suppose otherwise, let's suppose the number of past years has the order type of the non-positive integers, and let's enumerate the past years in the following way: [25]

Last year is 0, the previous year is -1, and so on.

For instance, if this year is 2012, then 2011 is 0, 2010 is -1, 2009 is -2, and so on.

Now, let F be a function from the set of non-positive integers into itself such that for all non-negative integers r and n, F(-r) = -n if and only if -n is the most recent year Shandy wrote about during the year -r.

For instance, if, in the year -2000, Shandy wrote about a day in the year -300001 and about a day in the year -300000, then F(-2000) = -300000.

Given the rate at which Shandy writes, and given also that, when writing his autobiography, Shandy never writes about his future, we have the following conditions:

1. F(-r) ≤ -r.

2. F(-r-365) = F(-r) - 1.

By induction:

3. F(-r-2*365) = F(-r-365-365) = F(-r-365) - 1=F(-r) - 2

4. F(-r-k*365) = F(-r) - k, for all non-negative k.

So, in particular, taking r=0.

F(0) - k = F(-k*365) ≤ -k*365.

Hence, for every natural number k,

5. 364*k ≤ -F(0)

That's contradictory, as easily seen by taking (for instance) k = 1 + (F(0)*F(0)).

So, the Tristram Shandy scenario fails to show that infinitely many past years are impossible.

In other words, what's logically impossible is Tristram Shandy scenario itself, but that does not have anything to do with whether an infinite past is possible.

Craig acknowledges that the scenario is logically impossible, yet claims that because it seems "obviously" coherent to write one's autobiography at a rate of one day per year, it seems to them that the problem is the infinite past.

I have to confess that I find his reply puzzling.

Clearly, the task of writing one's autobiography at a rate of one day per year 'from infinity' is logically impossible, and so not coherent. I honestly do not understand what else would be needed to reject the 'Tristram Shandy' scenario as an argument against an infinite past.

Incidentally, there is a contradictory scenario about the future that strikes me as similar to the 'Tristram Shandy' scenario:

Alice Shandy writes a novel starting in the year 2000, and the story is set in her future. She always writes about future days. Also, it takes Alice a year to account for what happens in one day in her novel, and she always writes her novel in sequence. In other words, she writes about what happens on some day d₁, and after she finishes writing about d₁, she moves to the day after d₁, without jumping to any later day.

Also, for any number n, Alice spends more than n years writing her novel.

The 'Alice Shandy' scenario is contradictory, but that does not warrant a conclusion that an unbounded future is metaphysically impossible.

Someone might insist that the task of writing a novel about the future, writing about consecutive days at a rate of one day per year, is obviously coherent, and so the fault must be with the idea of a future with an unbounded number of years, but it seems apparent to me that such a reply would be very mistaken.

The 'Tristram Shandy' scenario strikes me as similar, but in any case, I would say that we should in any case reject the 'Tristram Shandy' scenario because the task of writing one's biography at a rate of one day per year 'from infinity' is incoherent, and so the fact that assuming it's possible results in absurdities does not tell us anything about whether an infinite past in which the set of past years has the order type of the negative integers is metaphysically possible.

4.4.2. Orbits and parity.

Another argument against an infinite past defended by Craig [26] is based on orbits and the alleged parity of some numbers.

According to this argument, it's absurd that if Jupiter completes 2.5 orbits for each one Saturn completes, they would both have completed the same number of orbits, if they have been orbiting the Sun for an infinite number of years.

However, and leaving aside the fact that any planet, star, etc., only lasts for finitely many years in the actual world, the fact is that there appears to be no absurdity at all. As in the case of the Hilbert Hotel argument, the problem seems to be a confusion about the meaning of the words.

More specifically, that the number of orbits would be the same if they had been orbiting forever and there were an infinite past composed of denumerably many years merely means that there would be a bijection between the set of orbits completed by one of the planets, and the same set for the other planet, and that's as clear as the fact that, say, the set of natural numbers that are multiples of two has the same cardinality as the set of natural numbers that are multiples of five.

In addition to the previous argument, Craig also says that if a planet had been orbiting for eternity, the number of completed orbits would be both even and odd, and provides an unusual definition of 'even' and 'odd' that would apply to infinite cardinals, and under which infinite cardinals would turn out to be both even and odd.

However, that is not a problem, either, but rather, a consequence of Craig's unusual choice of definitions and the fact that there is a bijection between any two denumerable sets.

Of course, once again in reality planets don't last indefinitely, but that is not at all relevant.

4.5. Discrete time and infinite regress.

In addition to his 'grim reapers' argument, Alexander Pruss[1 8] gives another argument against infinite regress if time is discrete:

First, he argues that rigid discreteness is probably not an option, due to Zeno's Stadium argument, and relativistic considerations. I already commented on those matters earlier.

Then, he argues against infinite regress of events under Aristotelian discreteness, under the definition he suggests.[1 8]

So, Pruss reasoning is basically that if there were an infinite sequence of past events D(-n), for all natural n, such that D(-n) precedes D(-m) if n>m, then D(-n) could cause something at time 1/n, contradicting the hypothesis that time is discrete in the Aristotelian sense. According to Pruss, there is no reason to rule out all of those happening together.

But it seems to me that there may well be, and one possible such reason is precisely the fact that any consistent scenario in which time is discrete in the Aristotelian sense, and in which there is infinite temporal regress is a scenario in which it's not the case that the event D(-n) causes something at time 1/n.

In other words, what Pruss seems to be doing is going from the finite to the infinite case, including in the infinite case scenario some entity or entities with the causal power to bring about all of those things at time 1/n, and which exercise such powers successfully. But that's precisely not what happens in a consistent scenario in which time is discrete in the Aristotelian sense, and in which there is infinite temporal regress.

At this point, someone might ask questions like.

a. What would prevent the extension from the finite to the infinite case?

b. What would prevent those entities, if they exist, from exercising their powers?

I think that those would be the wrong questions, since it's not that something is preventing them from exercising powers. Rather, the problem is that one scenario is contradictory, but others aren't.

But purely for example, a consistent scenario in which time is discrete in the Aristotelian sense and in which there is infinite temporal regress and no entities with such powers may well be possible. Theists may believe otherwise as long as they believe in an all-powerful necessary entity or something like that, but I see no good reason to follow them on that.

For instance, the following scenario S₄ appears to be consistent:[27]

1. Time is discrete in the Aristotelian sense.

2. The set of years before some year – say, 0 – has the order type of the negative integers.

3. For any n > 10¹⁰⁰, there is no entity in the year -n with the power to bring about something in year 0, at time 1/n seconds (i.e., 1/n seconds after the year in question begins).

Granted, someone might add further conditions and reach a contradiction, but that's not the issue.

Also, someone might suggest that the 10¹⁰⁰restriction is arbitrary, but that would miss the point. For that matter, the following scenario S₅appears to be consistent as well:

1. Time is discrete in the Aristotelian sense.

2. The set of years before some year – say, 0 – has the order type of the negative integers.

3. For any natural number n > 10¹³⁹⁸³, there is no entity in the year -n with the power to bring about something in year 0, at time 1/n (i.e., 1/n seconds after the year in question begins).

4. For any natural number n < 10¹⁰⁰⁰, there is an entity in the year -n with the power to bring about something in year 0, at time 1/n seconds.

On the other hand, the following scenario S₆ (for instance) would be impossible:

1. Time is discrete in the Aristotelian sense.

2. The set of years before some year – say, 0 – has the order type of the negative integers.

3. For any natural number n, there is an entity in the year -n with the infallible power to bring about something in year 0, at time 1/n seconds (i.e., 1/n seconds after the year in question begins).

So, we can construct consistent scenarios like S₄orS₅, as well as inconsistent ones like S₆, but that does not seem to tell us anything about whether an infinite past is possible if time is discrete in the Aristotelian sense.

In any event, I would say that constructing logically contradictory scenarios like the one proposed by Pruss above does not seem to tell us anything about time, as far as I can tell.

All that said, if it turned out that an infinite temporal regress is indeed incompatible with time's being discrete in an Aristotelian sense, that would still not rule out infinite temporal regress, unless it's established that time is discrete in an Aristotelian sense.

Incidentally, this argument, if it worked, would seem to provide a blueprint for essentially similar arguments against, say, infinitely many unembodied beings (like angels), or generally infinitely many objects that can be programmed to carry out a certain tasks in the future, etc., under the assumption that time is discrete in an Aristotelian sense.

For instance, a theist who accepts Pruss' reasoning might also argue as follows: let's say that God can create infinitely many angels {A(n)} in a finite past, before the year 2013. Then, each angel A(n) could be created with the task of doing something on, say, December 25, 2113, at noon GMT + 1/n second, and that would contradict that time is discrete in an Aristotelian sense.

There are, of course, non-theist variants, and these kind of arguments might be deployed against all other sorts of infinities, and maybe even against the possibility of infinitely many concrete objects.

In any case, I find the argument unpersuasive even under the assumption that time is discrete in an Aristotelian sense, for the reasons given above.

4.6. Scientific cosmology.

According to Craig, modern scientific cosmology supports the second premise of the Kalam Cosmological Argument. In this section, I will consider his arguments on the matter.

4.6.1. The Friedmann–Lemaître Model.

Craig contends that what he calls the "Standard Hot Big Bang Model", or Friedmann-Lemaître model [15 ], supports a beginning of the universe that is an absolute origin ex-nihilo, and space and time themselves come into being at an initial singularity.

However, a singularity in that mathematical model indicates that the equations of General Relativity don't properly describe some phenomena in the distant past, not that there is some actual thing of infinite density – that would not even be defined.

In fact, General Relativity only takes into consideration one force – namely, gravity -, ignoring the rest, and while that's good enough for sufficiently large objects, in a very small universe like the universe in a distant past, other forces and the possibility of quantum effects should be taken into account as well. [29 ]

More precisely, what was small was the universe in the sense in which the word 'universe' is used in some scientific models, but whether the universe was small in the sense in which the word 'universe' is used in the KCA[14] is another matter.

In any event, the proper conclusion here is that we're not justified in applying those equations beyond a certain time in the past, which of course provides no support for the second premise of the KCA.

In addition to that, the model in question entails an infinite past sequence of changes, since as we move further back in time, the density of the universe tends to infinity, and so there are more than n consecutive changes in which the density of our universe decreases, for each n, even if the past in that model is metric-finite.

So, one of the models that Craig offers in support of his claims against an infinite past is incompatible with his stance on the possibility of actual infinities.

Granted, Craig or another defender of the KCA may simply point out some of what I mentioned above, namely that the model in question is not applicable beyond a certain time in the past, and so their stance on infinities is not undermined or defeated. But that only underscores the fact that this cosmological model is not applicable beyond a certain time in the past, and thus for that reason of no use for a defender of the KCA, even if the were 'universe' were being used in the relevant sense.

4.6.2. The Borde-Guth-Vilenkin Theorem.

In addition to the previous model, Craig often appeals[3 0 ] to a paper by Guth, Borde and Vilenkin[3 1 ], and offers that paper and the theorem proved in it as evidence in support of the second premise of the KCA.

In that paper, the authors reach the conclusion that if some reasonable assumptions obtain, then the past boundary of the inflating universe is reached in a finite past, and some new physics is required to described that boundary.

However, they make no claims in that paper about what the new physics would be, beyond mentioning that several alternatives have been discussed, including a quantum event in which the universe originates.

Granted, Vilenkin claims elsewhere that the universe had a beginning. I will address that claim later, but here I'd like to point out that if such quantum event actually happened a finite time ago, that would be part of the universe in the sense in which the word 'universe' is used in the KCA.[1 4 ]

Would that imply that the universe, in the sense in which the word 'universe' is used in the KCA, began to exist?

That would need to be argued for. Showing a beginning of the universe in a narrower sense of 'universe' would not suffice. I will address this matter in greater detail in the following subsection.

4.6.3. Other arguments based on scientific cosmology.

Craig offers a number of other arguments, based on different hypotheses in modern science, basically arguing that all of the hypotheses that aren't too implausible lead to the conclusion that the universe had a beginning.

A crucial issue here is what is meant by 'universe' in the context of those hypotheses, and what is meant by 'universe' in the context of the KCA. [1 4 ]

What a defender of the KCA would have to show in order to properly support premise 2 is not only that the object called 'universe' in the context of those scientific hypotheses began to exist, but that all of those hypotheses support a beginning of the universe in the sense of the word 'universe' that is relevant in the KCA. [3 2 ] In particular, they would still need to show that, plausibly, either:

i. There is no time in some realm that is prior to the existence of the object called 'universe' in those scientific hypotheses, or

ii. If there is time in some realm that is prior to the existence of the object called 'universe' in those scientific hypotheses, that realm would plausibly not qualify as 'universe' in the context of the KCA.

But Craig and other defenders of the KCA have not shown that, so the burden remains on them.

Still, we may speculate about what kind of argument would be required to support the second premise of the KCA using empirical evidence.

Perhaps, a defender of the KCA might try a probabilistic argument like the following argument:

If the object called 'universe' in scientific models had an infinite past, then so would the universe in the sense in which the word 'universe' is used in the KCA.

So, if, based on the empirical evidence available to us, we should revise our prior probabilistic assessment of the hypothesis that the object called 'universe' in the context of scientific models has an infinite past, assigning higher probability to that hypothesis than before the empirical evidence was considered, then we should also reduce the probability assigned to the second premise of the KCA.

In other words, empirical evidence that increases the probability that the object called 'universe' in the context of scientific models has an infinite past, also decreases the probability that the second premise of the KCA is true.

But then, reciprocally, empirical evidence that decreases the probability that the object called 'universe' in the context of scientific models has an infinite past, also increases the probability that the second premise of the KCA is true.

A first difficulty here is that different models might use the word 'universe' somewhat differently.

But let's assume that 'universe' means the same in all of the models, to simplify.
Even then, the probabilistic argument above would not be enough to establish that the second premise of the KCA is true, or even probably true.

In order to justify an assessment that the second premise of the KCA is probably true, based on empirical evidence, a defender of the KCA would also have to show that:

a. Based on empirical evidence, we should conclude that the object called 'universe' in scientific models probably has a finite past. He would not have to show what the probability P is, but show it's P > 0.5.

b. Given the probability P in a., it is probable that the universe, in the sense in which the word 'universe' is used in the KCA[1 4 ], has a finite past.

However, Craig and other defenders of the KCA have not even shown that, assuming P=1, then it's probable that the universe, in the sense in which the word 'universe' is used in the KCA, has a finite past.

Moreover, even if we granted for the sake of the argument that if P has a high value (say, P > 0.8), then b. is true, still Craig and other defenders of the KCA have not shown that we ought to assign such a high value to P. On that note, in the next subsection I will address the hypothesis of a cyclic universe with an infinite past, and Craig's reply to it.

4.6.4. Cyclic models and other options for an infinite past.

Some of the models of a universe with an infinite past that Craig considers and rejects[33 ] in his defense of the KCA are cyclic models of the universe.

For instance, in particular, he considers a model proposed by Frampton. [3 4 ]

In his assessment, Craig mentions some of the objections to the model and the replies by Frampton, and in the end concludes that the field is too young to make a full judgment, but that nevertheless it seems that some of the problems of older cyclic models remain.

So, even though Craig says that he's not passing full judgment, he does give the impression that he's making some kind of probabilistic assessment, implicitly saying that some of the objections that defeated some earlier cyclic models probably will defeat this one as well, and further, will defeat all cyclic models. But Craig does not provide any adequate support for such an assessment.

In any case, and regardless of Craig's claims about that particular cyclic model, a question in this context is whether we should assess that cyclic models of a universe with an infinite past probably fail. I do not see any sufficient reason to reach that conclusion at this point.

Granted, there are objections to such models, but no model, cyclic or otherwise, is generally accepted at this point. Rather, scientists are working on the development of different hypotheses, and a number of different options remain open.

In particular, the option of a cyclic universe with an infinite past continues to be a live one, and there are several recent papers discussing a number of variants of it as potential options.[3 5 ]

Moreover, even leaving aside cyclic models, there appear to be non-cyclic options of a universe with an infinite past still on the table.[3 6 ]

So, it is clear that whether the past is infinite is an open question in science, and there appears to be no good reason to take a stance at this point. In the next subsection, I will consider a recent disagreement between prominent cosmologists on the issue of an infinite past.

4.6.5. The Susskind-Vilenkin discussion.

In a recent paper, Vilenkin argues that the universe is not past eternal and has an initial condition in a finite past. [3 7 ] Susskind published another paper [38 ], in reply to Vilenkin's paper, and then another one after a discussion between Susskind, Vilenkin, and Bousso. [39 ]

Susskind disagrees with Vilenkin's arguments, and while he takes no stance on whether there is an initial condition, he argues that an inflating universe with an infinite future also has an infinite past. [4 0 ]

So, let's assume that it's been established that the universe undergoes inflation – which is accepted by Vilenkin -, and let's assume that the future will not be finite – which seems plausible -, and let's consider hypothesis.

Alternative 1: Vilenkin is right, and then there is a beginning in a finite past.

That would perhaps be the best case scenario for the defender of the second premise of the KCA. However, as I argued above, even if it's established that Vilenkin is right, in order to establish premise 2 of the KCA, it would have to established that there is a beginning of the universe in the sense of the word 'universe' used in the KCA. I do not claim that this is not doable, but rather, that the burden is on the defender of that premise.

Alternative 2. Susskind is right, and the probability that the universe existed over n years ago is 1, for all n.

In that case, there are two hypotheses compatible with Susskind's conclusion, in the sense that if Susskind is right, then the probability that at least one of the two following hypotheses obtains, is 1.

Alternative 2.a. There is no initial condition of the universe.

Then, it's clear that the universe did not have a beginning in the sense in which 'universe' is used in the exchange between Susskind and Vilenkin.

Thus, neither did the universe have a beginning in the sense in which the word 'universe' is used in the KCA, since the universe in the sense in which the word 'universe' is used in the KCA at least contains the universe in the sense in which the word 'universe' is used in science.

Hence, under this scenario, premise 2 of the KCA is false.

Alternative 2.b. There is an initial condition of the universe, but the initial condition is in the infinite past.

Using the expression 'begins to exist' in the sense in which Craig understands it, it seems plausible that the universe didn't begin to exist if it had an initial condition in the infinite past, since:

i. Craig imposes the condition of a tensed theory of time for something to begin to exist.

ii. While I don't see good reason s to conclude that a tensed theory of time is incompatible with a beginningless infinite past, it seems clear to me that a tensed theory of time is incompatible with an infinite past with a beginning point, since in that case, there would be states of the universe separated by infinitely many years, and it doesn't seem that that infinite gap between two given states could ever be traversed.

Still, perhaps, a different defender of the KCA might not impose the restriction of a tensed theory of time. On that note, under this second alternative, and using the expression 'begins to exist' intuitively – rather than as Craig understands it -, I would say that the universe did begin to exist in the infinite past, in the sense in which the word 'universe' is used in the exchange between Susskind and Vilenkin. [4 1 ]

However, in order to that premise 2 is true, a defender of the KCA would have to show that the universe in the sense in which the word 'universe' is used in the KCA[1 4 ], began to exist.

5. Conclusion.

Neither the first nor the second premise of the KCA is properly supported by the arguments given by Craig and others who propose them, and I see no good reason independently of those arguments to accept said premises, either. In particular, contemplating that the second premise might be true seems to undermine intuitive support for the first.

Additionally, there are very serious questions about the coherence of the creator and/or the creation account proposed by Craig, and generally by any similar versions of the KCA+. To my knowledge, there are no present-day versions of the KCA+ that avoid such difficulties.

6. Notes and References.

[1] Source: William Lane Craig and J. P. Sinclair, "The Kalam Cosmological Argument", in "The BlackWell Companion to Natural Theology", Edited by William Lane Craig and J. P. Moreland, © 2009 Blackwell Publishing Ltd. ISBN: 978-1-405-17657-6
Page 102.

[2] In some contexts, people might talk about, say, a timeless work of art, but that's clearly not the sense of 'timeless' that is relevant here.

[3] Source: http://www.reasonablefaith.org/site/News2?page=NewsArticle&id=5971

[4] Source: William Lane Craig and J. P. Sinclair, "The Kalam Cosmological Argument", in "The BlackWell Companion to Natural Theology", Edited by William Lane Craig and J. P. Moreland, © 2009 Blackwell Publishing Ltd. ISBN: 978-1-405-17657-6
Page 106.

[5] Source: http://www.reasonablefaith.org/site/News2?page=NewsArticle&id=9269

[6] Or the only concrete object is O, if abstracta are part of a correct ontology. That issue is not relevant to the argument under consideration, so we may stipulate anything in that regard if we so choose.

[7] It's not clear to me whether Craig uses 'changeless' and 'quiescent' to mean the same. But either way, the problems for his position remain, as argued above.

[8] Source: William Lane Craig and J. P. Sinclair, "The Kalam Cosmological Argument", in "The BlackWell Companion to Natural Theology", Edited by William Lane Craig and J. P. Moreland, © 2009 Blackwell Publishing Ltd. ISBN: 978-1-405-17657-6; pages 115, 116.

[9] According to Craig, "the universe" in the context of the KCA is defined as "the whole of material reality".

Source: http://www.reasonablefaith.org/causal-premiss-of-the-kalam-argument

It's not entirely clear to me what "material" means, but there is no need for greater clarification at this point. At any rate, we may just say that in this context, 'universe' means whatever it means in the second premise of the KCA.

[10] Source: William Lane Craig, "God and the Beginning of Time".

http://www.reasonablefaith.org/god-and-the-beginning-of-time

[11] Source: William Lane Craig and J. P. Sinclair, "The Kalam Cosmological Argument", in "The Blackwell Companion to Natural Theology", Edited by William lane Craig and J. P. Moreland; pages 184, 185.

[12] Sources: William Lane Craig and J. P. Sinclair, "The Kalam Cosmological Argument", in "The BlackWell Companion to Natural Theology", Edited by William Lane Craig and J. P. Moreland, © 2009 Blackwell Publishing Ltd. ISBN: 978-1-405-17657-6; page 182.

http://www.reasonablefaith.org/defenders-2-podcast/transcript/s4-6

[13] Our intuitions about space and time, while very useful in daily life, do not work so well in some of the situations modern physics deals with, like the proximity of a black hole, or the early universe, or the very small, etc. It's not at all clear, to say the least, that those intuitions are reliable in contexts like a purported beginning of the universe, etc.

However, given that defenders of the KCA propose intuitions in support of the first premise, we may offer rebuttals based on intuitions as well, among others.

[14] According to Craig, 'universe' means "the whole of material reality" in the context of the KCA.

Source: http://www.reasonablefaith.org/causal-premiss-of-the-kalam-argument

As I mentioned above, it's not clear to me, however, what "material" means, though it seems clear that if there were a multiverse, infinite past cyclic universes, etc., all of that would be part of the universe in the intended sense.

[15] Source: William Lane Craig and J. P. Sinclair, "The Kalam Cosmological Argument", in "The BlackWell Companion to Natural Theology", Edited by William Lane Craig and J. P. Moreland, © 2009 Blackwell Publishing Ltd. ISBN: 978-1-405-17657-6; page 108-115.

[16] That would be debatable on presentism, which is precisely the theory of time that Craig espouses. But let's grant for the sake of the argument that no actual infinity entails no infinite past, even on presentism.

[17] Greater infinities would not make a relevant difference, so one may assume a larger infinity for that matter, and essentially the same counterarguments would work just as well.

[18] Sources:

http://prosblogion.ektopos.com/archives/2009/10/from-grim-reape.html

http://alexanderpruss.blogspot.com/2009/10/from-grim-reaper-paradox-to-kalaam.html

More details on the definition of Aristotelian discreteness suggested by Pruss: http://prosblogion.ektopos.com/archives/2009/10/from-grim-reape.html#comment-107238

While Pruss' does not seem to explicitly say that, under Aristotelian discreteness, it's necessary that there are only finitely many moments or intervals between two given times.

However, it's clear in context that his conception of time that is discrete in an Aristotelian sense rules out that time is possibly dense. On that note, he gives an argument against infinite regress assuming that time is Aristotelian discrete, and in which he concludes that the mere possibility of temporal density would contradict the assumption that time is discrete in an Aristotelian sense.

[19] Someone might object to the expression "moments of time", holding that there are only intervals, not instants. That's not a crucial matter in the context of this part of the argument, and the same kind of argumentation can be given assuming that there are only intervals, changing the definitions as needed.

[20] Source: Alexander Pruss, in "The Leibnizian Cosmological Argument", in "The BlackWell Companion to Natural Theology", Edited by William Lane Craig and J. P. Moreland, © 2009 Blackwell Publishing Ltd. ISBN: 978-1-405-17657-6.

Pruss' defense of Leibnizian arguments can be found in the following page: https://bearspace.baylor.edu/Alexander_Pruss/www/papers/LCA.html

[21] Sources: Koons, R. C. (2012), A New Kalam Argument: Revenge of the Grim Reaper. Noûs. doi: 10.1111/j.1468-0068.2012.00858.x

http://www.robkoons.net/media/83c9b25c56d629ffffff810fffffd524.pdf

[22] Koons uses the word "set", but it's not entirely clear to me that that would be a set.

For instance, for all I know, there might be a possible world W with physical structures of cardinality x, for any cardinal x. If so, the class of spatio-temporal regions of W would appear not to be a set.

So, I prefer to use the word 'class'. In any case, this is a side issue.

[23] Source: William Lane Craig and J. P. Sinclair, "The Kalam Cosmological Argument", in "The BlackWell Companion to Natural Theology", Edited by William Lane Craig and J. P. Moreland, © 2009 Blackwell Publishing Ltd. ISBN: 978-1-405-17657-6; page 117-124.

[24] Josh Dever, in "Worlds Apart", Taiwanese Journal for Philosophy and History of Science, 10 (1998), pointed out that the scenario is contradictory.

This was also pointed out by Graham Oppy, in "Arguing about Gods", Cambridge University Press (2006).

[25] I'm stipulating 1 year = 365 days and ignoring leap years, for the sake of simplicity; a more complicated proof would include leap years, but it's clear that the contradiction does not depend on whether we take into consideration leap years.

Also, the assumption that the set of past years has the order type of the non-positive integers is acceptable in this context, since that is the main hypothesis that Craig claims ought to be rejected due to the Tristram Shandy argument, and since in any case, I intend to show that the 'Tristram Shandy' argument does not show that such a past is impossible.

[26] Source: William Lane Craig and J. P. Sinclair, "The Kalam Cosmological Argument", in "The BlackWell Companion to Natural Theology", Edited by William Lane Craig and J. P. Moreland, © 2009 Blackwell Publishing Ltd. ISBN: 978-1-405-17657-6; page 120.

[27] As usual, the scenario isn't completely specified, so someone might argue that it's more accurate to say it's a type or category of scenario, but calling them 'scenario' seems to be pretty standard. In any case, that they're not fully specified shouldn't cause any confusion, as that's normally the case.

[28] Here, B(n) may or may not be the same being for all n; that's not important in this context.

[29] Sources:

1. Matts Roos, "Expansion of the Universe – Standard Big Bang Model". http://arxiv.org/abs/0802.2005

2. http://csep10.phys.utk.edu/astr162/lect/cosmology/planck.html

3. http://www.nicadd.niu.edu/~bterzic/PHYS652/Lecture_13.pdf

4. http://preposterousuniverse.com/writings/dtung/

[30] Sources (for example):

William Lane Craig and J. P. Sinclair, "The Kalam Cosmological Argument", in "The BlackWell Companion to Natural Theology", Edited by William Lane Craig and J. P. Moreland, © 2009 Blackwell Publishing Ltd. ISBN: 978-1-405-17657-6; page 141.

http://www.reasonablefaith.org/contemporary-cosmology-and-the-beginning-of-the-universe

[31] Source: Guth, Borde, and Vilenkin, "Inflationary spacetimes are not past-complete", Physical

Review Letters 90, 151301.

http://arxiv.org/abs/gr-qc/0110012

[32] I'm taking into consideration the fact that defenders of the KCA are actually also defenders of the KCA+, and that rules out the use of the word 'universe' in the KCA in a narrow sense, such as the use that it might be given to the word 'universe' in some scientific models.

At any rate, using 'universe' in a narrow sense in the second premise of the KCA would on its own make the argument irrelevant in the context of philosophy of religion, since that conclusion would not rule out the that there was some older realm before the universe, and more precisely a realm that does not entail or suggest a personal creator.

[33] Source: William Lane Craig and J. P. Sinclair, "The Kalam Cosmological Argument", in "The BlackWell Companion to Natural Theology", Edited by William Lane Craig and J. P. Moreland, © 2009 Blackwell Publishing Ltd. ISBN: 978-1-405-17657-6; pages 150-157.

[34] Paul Frampton, "Cyclic Universe and Infinite Past".

http://arxiv.org/abs/0705.2730

[35] There are many examples. I will mention a few, but one might as well choose others.

To be clear, I'm not suggesting that any particular model is likely, but merely pointing out that the matter of an infinite cyclic past is open in modern scientific cosmology.

So, for example, one can mention – among others – the following papers:

Yun-Song Piao, "Proliferation in Cycle". http://arxiv.org/abs/0901.2644

Yun-Song Piao, "Design of a Cyclic Multiverse". http://arxiv.org/abs/1001.0631

Hua-Hui Xiong, Yi-Fu Cai, Taotao Qiu, Yun-Song Piao, Xinmin Zhang, "Oscillating universe with quintom matter". http://arxiv.org/abs/0805.0413

Kazuharu Bamba, Kuralay Yesmakhanova, Koblandy Yerzhanov, Ratbay Myrzakulov, "Reconstruction of the equation of state for the cyclic universes in homogeneous and isotropic cosmology". http://arxiv.org/abs/1203.3401v2

Yi-Fu Cai, Emmanuel N. Saridakis, "Cyclic cosmology from Lagrange-multiplier modified gravity". http://arxiv.org/abs/1007.3204

Kazuharu Bamba, Ujjal Debnath, Kuralay Yesmakhanova, Petr Tsyba, Gulgasyl Nugmanova, Ratbay Myrzakulov, "Periodic Cosmological Evolutions of Equation of State for Dark Energy". http://arxiv.org/abs/1203.4226.

[36] For instance:

Pierre-Henri Chavanis, "A simple model of universe with a polytropic equation of state"- http://arxiv.org/abs/1208.1192

[37] Audrey Mithani, Alexander Vilenkin, "Did the universe have a beginning?". http://arxiv.org/abs/1204.4658

In fact, Vilenkin and Mithani argue that eternal inflation, cyclic universes, and an emergent universe all result in a finite past, given certain plausible conditions. I already mentioned alternative cyclic models with an infinite past, so those models implicitly disagree with Vilenkin and Mithani's conclusions.

Susskind makes a different argument. He actually rejects cyclic models as well, but makes an argument under a universe that undergoes inflation – Vilenkin accepts a universe with inflation, but disagrees with Susskind on the conclusions about the past -, reaching a conclusion that if it has an infinite future [which is plausible], then with probability 1, it has an infinite past.

[38] Leonard Susskind, "Was There a Beginning?"

http://arxiv.org/abs/1204.5385

[39] Leonard Susskind, "Is Eternal Inflation Past-Eternal? And What if It Is?". http://arxiv.org/abs/1205.0589

[40] A supporter of the KCA might point out that Susskind rejects cyclic beginningless universes. That is true, but not relevant to the point I'm making here. I'm offering the exchange between Susskind and Vilenkin as an example of how contentious the matter of an infinite past still is, among present-day experts, not taking a stance in favor of Susskind, or suggesting that he should be accepted as an undisputed authority. On the contrary, disagreement and discussions are common on this matter.

[41] However, I would still see no good reason to think that, in that scenario, the universe in that sense came into existence, or that it had a cause, unless there was time before that point in the infinite past. For the reasons I gave earlier, I do not think that premise 1 of the KCA is properly supported.

Angra Mainyu's blog

Pages

Sunday, January 6, 2013

A Reply to the Kalam Cosmological Argument (v 2.3)